Challenges In Social Network Research: Methods And Applications 3030314626, 9783030314620, 9783030314637

Table of contents :
Preface......Page 6
Contents......Page 9
Contributors......Page 11
1.1 Network Element Importance Measures......Page 14
2.1 Overlap Weight......Page 15
2.2 US Airports Links with the Largest Overlap Weight......Page 18
2.3 Corrected Overlap Weight......Page 20
2.4 US Airports 1997 Links with the Largest Corrected Overlap Weight......Page 21
3.1 Clustering Coefficient......Page 22
3.2 US Airports with the Largest Clustering Coefficient......Page 24
3.3 Corrected Clustering Coefficient......Page 25
3.5 Comparisons......Page 26
4 Conclusions......Page 27
References......Page 29
1 Introduction......Page 30
2 Politics, Analysis of Multilevel Networks, and Multilevel Relational Infrastructures......Page 32
3 Bottom-Up Collegiality, Top-Down Collegiality, and Inside-Out Collegiality......Page 34
4 An Example of Top-Down Collegiality in Institutional Entrepreneurship......Page 36
5 The Challenge of Contextualizing Multilevel Networks: Organized Mobility and Relational Turnover......Page 38
6 Multispin for Contextualizing Multilevel Networks......Page 39
7 Conclusion......Page 41
References......Page 42
Part I Methods......Page 45
1 Introduction......Page 46
2 Related Work......Page 47
3 Method......Page 48
4 Empirical Results......Page 51
4.1 Hatfield–McCoy Case Study......Page 52
4.2 Ukrainian Parliament Case Study......Page 54
5 Discussion......Page 56
6 Conclusion and Future Work......Page 57
References......Page 58
1 Introduction......Page 60
2 The Approach......Page 61
3 Simulations......Page 62
4 Application......Page 67
5 Conclusions......Page 71
References......Page 72
1 Introduction......Page 74
2 Theoretical Background......Page 75
3 The Data......Page 77
4 The Analytic Strategy......Page 78
4.1 Association Rules Mining the Prescriptions Dataset......Page 80
5 The Results......Page 82
5.1 First Network Results......Page 83
6 Discussion and Concluding Remarks......Page 84
References......Page 88
1 Introduction......Page 90
2 Model-Based Clustering for Bipartite Networks......Page 91
3 Noordin Top Terrorist Network......Page 93
3.1 Statistical Analysis......Page 94
3.2 Interpreting the Actor's Behaviour......Page 95
3.3 Interpreting the Events Attendance......Page 96
References......Page 101
1 Introduction......Page 103
2 Related Works......Page 104
3 A DEA-Based Network Formation Model......Page 106
4.1 Macro Analysis......Page 111
4.2 Micro Analysis......Page 113
5.1 Simulation......Page 114
6 Discussion and Conclusions......Page 119
A.1 Appendix 1 DEA Analysis of Peers......Page 120
B.1 Appendix 2 Adjacency Matrix for DEA-Based Network Representation......Page 121
References......Page 122
1 Context-Aware Social Network Research......Page 125
2 Definition of Context......Page 127
3 Contextualized Networks......Page 128
3.2 Context in Document Networks......Page 129
4 Analytical Potential and Related Work......Page 130
4.2 Network-Level Metrics......Page 131
4.4 Community Detection......Page 132
4.5 Visualizing Networks......Page 133
5 Algorithmic Challenges......Page 134
5.2 Organization of Combined Structural and Context Data......Page 135
5.4 Linking to Raw Data......Page 136
6 Conclusion......Page 137
References......Page 138
Part II Applications......Page 141
Unraveling Innovation Networks in Conservation Agriculture Using Social Network Analysis......Page 142
1 Introduction......Page 143
The Affiliation Network......Page 145
Network of Events (Innovations)......Page 146
2.4 Data Analysis......Page 147
3.1 Actors and CA Practices......Page 148
3.2 Pattern of Agricultural Practices......Page 149
3.3 Innovation Adoption Based on Network and Individual Attributes......Page 151
4 Conclusions......Page 154
References......Page 155
Mapping Informal Organization Through Urban Activism: The Case of Self-Organized Spaces in the City of Naples......Page 158
1 Introduction......Page 159
2 Research Strategy and Method......Page 160
3 Data Mining and Visualization......Page 164
4 Evidence......Page 170
5 Conclusions......Page 174
References......Page 175
1 Introduction......Page 179
2 Italian Poetic Context......Page 181
3 Methods and Procedures......Page 183
4 Results and Discussion......Page 185
4.1 The Poets' Affiliation Network......Page 186
4.2 A Multivariate Approach for Studying Poetic Production......Page 194
5 Conclusions......Page 195
References......Page 197
Multilayer Network Analysis of Innovation Intermediaries' Activities......Page 200
1 Issue: Intermediaries in Innovation Processes......Page 201
2.1 The Regional Policy......Page 203
2.3 Structuring the Data as a Multilayer Network......Page 204
3.1 Flows of Information in a Multilayer Network Perspective......Page 207
3.2 Settings of the Algorithm......Page 208
4 Main Results......Page 209
5 Lessons from the Multilayer Analysis......Page 211
References......Page 214
1 Introduction......Page 216
2 The Network Perspective in the Analysis of Inter-Organizational Systems......Page 217
3 Networks and Welfare Systems in Italy: Some Empirical Findings......Page 219
4 The Collaboration Networks of the Caritas in Aversa and Benevento......Page 221
5 Assessing Inter-Organizational Networks......Page 222
6 The Research Plan: Ego-Whole Network Construction, Data Collection and Analysis......Page 224
7.1 The Structure of Networks: Relational Embeddedness and Structural Embeddedness......Page 226
7.2 Structural Equivalence and Blockmodelling for the Analysis of Similarity......Page 232
8 Discussion and Conclusions......Page 235
References......Page 238
Index......Page 241

Citation preview

Lecture Notes in Social Networks

Giancarlo Ragozini Maria Prosperina Vitale   Editors

Challenges in Social Network Research Methods and Applications

Lecture Notes in Social Networks Series editors Reda Alhajj, University of Calgary, Calgary, AB, Canada Uwe Glässer, Simon Fraser University, Burnaby, BC, Canada Huan Liu, Arizona State University, Tempe, AZ, USA Rafael Wittek, University of Groningen, Groningen, The Netherlands Daniel Zeng, University of Arizona, Tucson, AZ, USA Advisory Board Charu C. Aggarwal, Yorktown Heights, NY, USA Patricia L. Brantingham, Simon Fraser University, Burnaby, BC, Canada Thilo Gross, University of Bristol, Bristol, UK Jiawei Han, University of Illinois at Urbana-Champaign, Urbana, IL, USA Raúl Manásevich, University of Chile, Santiago, Chile Anthony J. Masys, University of Leicester, Ottawa, ON, Canada Carlo Morselli, School of Criminology, Montreal, QC, Canada

Lecture Notes in Social Networks (LNSN) comprises volumes covering the theory, foundations and applications of the new emerging multidisciplinary field of social networks analysis and mining. LNSN publishes peer-reviewed works (including monographs, edited works) in the analytical, technical as well as the organizational side of social computing, social networks, network sciences, graph theory, sociology, Semantics Web, Web applications and analytics, information networks, theoretical physics, modeling, security, crisis and risk management, and other related disciplines. The volumes are guest-edited by experts in a specific domain. This series is indexed by DBLP. Springer and the Series Editors welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Christoph Baumann, Publishing Editor, Springer e-mail: [email protected]

More information about this series at http://www.springer.com/series/8768

Giancarlo Ragozini • Maria Prosperina Vitale Editors

Challenges in Social Network Research Methods and Applications

123

Editors Giancarlo Ragozini Department of Political Science University of Naples Federico II Naples, Italy

Maria Prosperina Vitale Department of Political and Social Studies University of Salerno Fisciano (SA), Italy

ISSN 2190-5428 ISSN 2190-5436 (electronic) Lecture Notes in Social Networks ISBN 978-3-030-31462-0 ISBN 978-3-030-31463-7 (eBook) https://doi.org/10.1007/978-3-030-31463-7 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The contributions included in this volume mainly stem from a selection of original papers presented at the Sixth International Workshop on Social Network Analysis, ARS’17, “Challenges in Social network research,” held in Naples (Italy) on May 16–17, 2017. The workshop was the sixth of a successful series. It aimed at presenting the most relevant results and the most recent methodological developments in social network research in order to broaden the knowledge of network analysis and discuss “where” and “how” this methodological perspective ought to be applied both in practice and in theory. ARS (“Analisi delle Reti Sociali”, i.e., “Social Network Analysis” in Italian) is a multidisciplinary group of scholars promoting research on social networks by organizing meetings, summer schools, and biennial workshops hosted by Italian universities, as well as by publishing dedicated special issues on the topic since 2007 celebrating the 10-year Edition in 2017. The book includes both invited and contributed chapters dealing with advanced methods and theoretical development for the analysis of social networks and applications in numerous disciplines. Some authors explore new trends related to network measures, multilevel networks, and clustering on networks, while other contributions deepen the relationship among statistical methods for data mining and social network analysis. Along with the new methodological developments, the book offers interesting applications to a wide set of fields, ranging from the organizational and economic studies, collaboration, and innovation to the less usual field of poetry. In addition, the case studies are related to local context, showing how the substantive reasoning is fundamental in social network analysis. The list of authors includes both top scholars in the field of social networks and promising young researchers. More specifically, the volume is introduced by two invited chapters. In the first chapter, V. Batagelj addresses the issues related to two network measures: the overlap weight of an edge and the clustering coefficient of a node proposing a corrected measure of both. The second chapter by E. Lazega presents a theoretical

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contribution on the role of the analysis of multilevel networks in the fields of politics, institutional entrepreneurship, and social change. The methodological and theoretical parts are opened by a chapter that discusses a novel methodology meant to derive networks and communities from socio-cultural data, based on socio-cultural cognitive mapping and k-NN network modularity maximization, authored by I. Cruickshank and K.M. Carley. The fourth chapter by C. Drago and R. Ricciuti presents a new method for the detection of changes in the network structure by a bootstrap test on the degree distribution. The analytical procedure is illustrated by using simulated data and a real-world example on interlocking directorship network. A two-step strategy of analysis is discussed in the chapter by G. Giordano et al. for studying comorbidity patterns exploiting association rules extracted by two-mode networks of prescriptions by pathologies and partitioning algorithms to identify the most relevant and connected parts in the one-mode network of pathologies. The sixth chapter by I. Gollini investigates the latent structure of bipartite networks via a model-based clustering approach. The model is exploited to identify the latent groups of terrorists and their latent trait scores based on their attendance to certain events. The chapter by C. Pinto proposes a network formation model based on Data Envelopment Analysis in which the usefulness of the agents depends exclusively on direct links and their formation. M. Schoenfeld and J. Pfeffer emphasize, in the eighth chapter, how network analysis benefits from considering context information and they identify the key challenges that need to be tackled from an algorithmic perspective. Along with the theoretical and methodological contributions, this volume also offers applications in: conservation agriculture and sustainability, urban community hubs, cultural network and market, policy design and evaluation, Third Sector and inter-organizational networks. In the ninth chapter, J. M. Aguirre-López et al. analyze the adoption patterns of conservation agriculture practices among a sample of maize smallholder farmers in the Mexican state of Chiapas. In the tenth chapter, after a preliminary investigation aimed at collecting a list of active organizations in Naples, Napolitano et al. verify and map the correspondence of activities performed by organizations with the priority themes of the cultural production system defined by the European Urban Agenda. In the eleventh chapter, S. Pedrini and C. Felaco explore the social dynamics and the interactions between young poets and the Italian poetic network with their cultural path by adopting Multiple Correspondence Analysis for twomode networks. In the twelfth chapter by M. Russo et al., the authors analyze the extent to which innovation intermediaries support the creation of communities of other agents through their engagement in different activities. Multilayer network analysis techniques are adopted to simultaneously represent the several types of interactions promoted by intermediaries. The book is closed by A. Salvini et al., who provide insights on the network governance of the social interacting organizations, presenting an empirical study of two networks of organizations operating in local territories in Southern Italy and focusing on Third Sector and welfare activities.

Preface

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The editors would like to dedicate this book to professors Maria Rosaria D’Esposito and Anuška Ferligoj, who have been the soul of the ARS group and our personal inspiring mentors in the research activities in this field over our whole academic life and are going to lead us in the future years to come as well. We would also like to thank the members of the Scientific Program Committee for contributing to make our conference successful. Special thanks are also due to the anonymous reviewers who helped us in the selection process of the papers. Naples, Italy Fisciano, Italy

Giancarlo Ragozini Maria Prosperina Vitale

Contents

Corrected Overlap Weight and Clustering Coefficient . . . . . . . . . . . . . . . . . . . . . . Vladimir Batagelj Bottom-Up Collegiality, Top-Down Collegiality, or Inside-Out Collegiality? Analyses of Multilevel Networks, Institutional Entrepreneurship and Laboratories for Social Change . . . . . . . . . . . . . . . . . . . . . Emmanuel Lazega

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Part I Methods Socio-Cultural Cognitive Mapping to Identify Communities and Latent Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iain Cruickshank and Kathleen M. Carley

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Bootstrapping the Gini Index of the Network Degree: An Application for Italian Corporate Governance . . . . . . . . . . . . . . . . . . . . . . . . . . . Carlo Drago and Roberto Ricciuti

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Association Rules and Network Analysis for Exploring Comorbidity Patterns in Health Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Giuseppe Giordano, Mario De Santis, Sergio Pagano, Giancarlo Ragozini, Maria Prosperina Vitale, and Pierpaolo Cavallo

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A Mixture Model Approach for Clustering Bipartite Networks . . . . . . . . . . . . Isabella Gollini

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A DEA-Based Network Formation Model. Micro and Macro Analysis . . . . Claudio Pinto

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Networks and Context: Algorithmic Challenges for Context-Aware Social Network Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Mirco Schoenfeld and Juergen Pfeffer

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Contents

Part II Applications Unraveling Innovation Networks in Conservation Agriculture Using Social Network Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Juan Manuel Aguirre-López, Julio Díaz-José, Petra Chaloupková, and Francisco Guevara-Hernández Mapping Informal Organization Through Urban Activism: The Case of Self-Organized Spaces in the City of Naples . . . . . . . . . . . . . . . . . . . 149 Pasquale Napolitano, Pierluigi Vitale, and Rita Lisa Vella The Paths of the Italian Young Poets. A Social Network Analysis of the Contemporary Poetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Sabrina Pedrini and Cristiano Felaco Multilayer Network Analysis of Innovation Intermediaries’ Activities . . . . 193 Margherita Russo, Annalisa Caloffi, Riccardo Righi, Simone Righi, and Federica Rossi Inter-Organizational Networks and Third Sector: Emerging Features from Two Case Studies in Southern Italy . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Andrea Salvini, Antonietta Riccardo, Francesco Vasca, and Irene Psaroudakis Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Contributors

Juan Manuel Aguirre-López Universidad Autónoma Chapingo, Chapingo, Mexico Vladimir Batagelj Department of Theoretical Computer Science, Institute of Mathematics, Physics and Mechanics, Jadranska, Ljubljana, Slovenia University of Primorska, Andrej Marušiˇc Institute, Koper, Slovenia National Research University Higher School of Economics, Moscow, Russia Annalisa Caloffi Department of Economics and Management, University of Florence, Florence, Italy Kathleen M. Carley Center for Computational Analysis of Social and Organizational Systems (CASOS), Institute for Software Research, Carnegie Mellon University, Pittsburgh, PA, USA Pierpaolo Cavallo Department of Physics E.R. Caianiello, University of Salerno, Fisciano (SA), Italy Petra Chaloupková Czech University of Life Sciences, Prague, Czech Republic Iain Cruickshank Center for Computational Analysis of Social and Organizational Systems (CASOS), Institute for Software Research, Carnegie Mellon University, Pittsburgh, PA, USA Mario De Santis Cooperativa Medi Service, Salerno, Italy Julio Díaz-José Tecnológico Nacional de México-Campus Zongolica, Veracruz, Mexico Facultad de Ciencias Biológicas y Agropecuarias, Universidad Veracruzana, Veracruz, Mexico Carlo Drago Niccolò Cusano University, Rome, Italy Cristiano Felaco University of Naples Federico II, Naples, Italy

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Contributors

Giuseppe Giordano Department of Political and Social Studies, University of Salerno, Fisciano (SA), Italy Isabella Gollini University College Dublin, Belfield, Dublin, Ireland Francisco Guevara-Hernández Universidad Autónoma de Chiapas, Villaflores, Chiapas, México Emmanuel Lazega Sciences Po, CSO-CNRS, IUF, Paris, France Pasquale Napolitano Science and Technology for Information and Communication Society, Graphics, Vision, Multimedia IRISS/CNR, Naples, Italy Sergio Pagano Department of Physics E.R. Caianiello, University of Salerno, Fisciano (SA), Italy Sabrina Pedrini University of Bologna Alma Mater Studiorum, Bologna, Italy Juergen Pfeffer Bavarian School of Public Policy, Technical University in Munich, Munich, Germany Claudio Pinto University of Salerno, Fisciano (SA), Italy Irene Psaroudakis University of Pisa, Pisa, Italy Giancarlo Ragozini Department of Political Science, University of Naples Federico II, Naples, Italy Antonietta Riccardo University of Pisa, Pisa, Italy Roberto Ricciuti University of Verona, Verona, Italy Riccardo Righi European Commission, Joint Research Centre (JRC), Seville, Spain Simone Righi Department of Computer Science, University College London, London, UK “Lendület” Research Center for Education and Network Studies (RECENS), Hungarian Academy of Sciences, Budapest, Hungary Federica Rossi Birkbeck, University of London, London, UK Margherita Russo Department of Economics, University of Modena and Reggio Emilia, Modena, Italy Andrea Salvini University of Pisa, Pisa, Italy Mirco Schoenfeld Bavarian School of Public Policy, Technical University in Munich, Munich, Germany Francesco Vasca University of Sannio, Benevento, Italy Rita Lisa Vella Department of Business and Management, LUISS Guido Carli, Rome, Italy

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Maria Prosperina Vitale Department of Political and Social Studies, University of Salerno, Fisciano (SA), Italy Pierluigi Vitale Department of Political and Communication Sciences, University of Salerno, Fisciano (SA), Italy

Corrected Overlap Weight and Clustering Coefficient Vladimir Batagelj

Abstract We discuss two well-known network measures: the overlap weight of an edge and the clustering coefficient of a node. For both of them it turns out that they are not very useful for data analytic task to identify important elements (nodes or links) of a given network. The reason for this is that they attain their largest values on maximal subgraphs of relatively small size that are more probable to appear in a network than that of larger size. We show how the definitions of these measures can be corrected in such a way that they give the expected results. We illustrate the proposed corrected measures by applying them to the US Airports network using the program Pajek. Keywords Social network analysis · Importance measure · Triangular weight · Overlap weight · Clustering coefficient

1 Introduction 1.1 Network Element Importance Measures To identify important/interesting elements (nodes, links) in a network we often try to express our intuition about their importance using an appropriate measure (node index, link weight) following the scheme: Larger is the measure value of an element, more important/interesting is this element.

Mathematics Subject Classification 2010: 91D30, 91C05, 05C85, 68R10, 05C42 V. Batagelj () Department of Theoretical Computer Science, Institute of Mathematics, Physics and Mechanics, Jadranska, Ljubljana, Slovenia University of Primorska, Andrej Marušiˇc Institute, Koper, Slovenia National Research University Higher School of Economics, Moscow, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. Ragozini, M. P. Vitale (eds.), Challenges in Social Network Research, Lecture Notes in Social Networks, https://doi.org/10.1007/978-3-030-31463-7_1

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V. Batagelj

Too often, in the analysis of networks, researchers uncritically pick some measure from the literature (degrees, closeness, betweenness, hubs and authorities, clustering coefficient, etc. [1, 2]) and apply it to their network. In this paper we discuss two well-known network local density measures: the overlap weight of an edge [3] and the clustering coefficient of a node [4, 5]. For both of them it turns out that they are not very useful for data analytic task to identify important elements of a given network. The reason for this is that they attain their largest values on maximal subgraphs of relatively small size—they are more probable to appear in a network than that of larger size. We show how their definitions can be corrected in such a way that they give the expected results. We illustrate the proposed corrected measures by applying them to the US Airports network using the program Pajek. We will limit our attention to undirected simple graphs G = (V, E). Many similar indices and weights were proposed by graph drawing community for disentanglement in the visualization of hairball networks [6–8]. When searching for important subnetworks in a given network we often assume a model that in the evolution of the network the increased activities in a part of the network create new nodes and edges in that part increasing its local density. We expect from a local density measure ld(x, G) for an element (node/link) x of network G the following properties: ld1. adding an edge, e, to the local neighborhood, G(1) , does not decrease the local density ld(x, G) ≤ ld(x, G ∪ e). ld2. normalization: 0 ≤ ld(x, G) ≤ 1. ld3. ld(x, G) can attain value 1, ld(x, G) = 1, on the largest subnetwork of certain type in the network.

2 Overlap Weight 2.1 Overlap Weight A direct measure of the overlap of an edge e = (u : v) ∈ E in an undirected simple graph G = (V, E) is the number of common neighbors of its end nodes u and v (see Fig. 1). It is equal to t (e)—the number of triangles (cycles of length 3) to which the edge e belongs. The edge neighbors subgraph is labeled T (deg(u) − t (e)−1, t (e), deg(v)−t (e)−1)—the subgraph in Fig. 1 is labeled T (4, 5, 3). There are two problems with this measure: • it is not normalized (bounded to [0, 1]); • it does not consider the “potentiality” of nodes u and v to form triangles—there are min(deg(u), deg(v)) − 1 − t (e) nodes in the smaller set of neighbors that are not in the other set of neighbors.

Corrected Overlap Weight and Clustering Coefficient

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Fig. 1 Neighbors of e(u : v)

u

e

v

Two simple normalizations are t (e) n−2

or

t (e) , μ

where n = |V| is the number of nodes, and μ = maxe∈E t (e) is the maximum number of triangles on an edge in the graph G. The (topological) overlap weight of an edge e = (u : v) ∈ E considers also the degrees of edge’s end nodes and is defined as o(e) =

t (e) . (deg(u) − 1) + (deg(v) − 1) − t (e)

In the case deg(u) = deg(v) = 1 we set o(e) = 0. It somehow resolves both problems. The overlap weight is essentially a Jaccard similarity index [9] J (X, Y ) =

|X ∩ Y | |X ∪ Y |

for X = N (u) \ {v} and Y = N(v) \ {u}, where N(z) is the set of neighbors of a node z. In this case we have |X ∩ Y | = t (e) and |X ∪ Y | = |X| + |Y | − |X ∩ Y | = (deg(u) − 1) + (deg(v) − 1) − t (e). | Note also that h(X, Y ) = 1 − J (X, Y ) = |X⊕Y |X∪Y | is the normalized Hamming distance [9]. The operation ⊕ denotes the symmetric difference X ⊕ Y = (X ∪ Y )\ (X ∩ Y ).

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V. Batagelj

Another normalized overlap measure is the overlap index [9] O(e) = O(X, Y ) =

|X ∩ Y | t (e) = . max(|X|, |Y |) max(deg(u), deg(v)) − 1

Both measures J and O, applied to networks, have some nice properties. For example: a pair of nodes u and v are structurally equivalent iff J (X, Y ) = O(X, Y ) = 1. Therefore the overlap weight measures the substitutiability of one edge’s end node by the other. Introducing two auxiliary quantities m(e) = min(deg(u), deg(v)) − 1 and

M(e) = max(deg(u), deg(v)) − 1

we can rewrite the definition of the overlap weight o(e) =

t (e) , m(e) + M(e) − t (e)

M(e) > 0

and if M(e) = 0, then o(e) = 0. For every edge e ∈ E it holds 0 ≤ t (e) ≤ m(e) ≤ M(e). Therefore m(e) + M(e) − t (e) ≥ t (e) + t (e) − t (e) = t (e) showing that 0 ≤ o(e) ≤ 1. The value o(e) = 1 is attained exactly in the case when M(e) = t (e), and the value o(e) = 0 exactly when t (e) = 0. In simple directed graphs without loops different types of triangles exist over an arc a(u, v). We can define overlap weights for each type. For example: the transitive overlap weight ot (a) =

tt (a) (outdeg(u) − 1) + (indeg(v) − 1) − tt (a)

and the cyclic overlap weight oc (a) =

tc (a) , indeg(u) + outdeg(v) − tc (a)

where tt (a) and tc (a) are the number of transitive/cyclic triangles containing the arc a. In this paper we will limit our discussion to overlap weights in undirected graphs.

Corrected Overlap Weight and Clustering Coefficient

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2.2 US Airports Links with the Largest Overlap Weight Let us apply the overlap weight to the network of US Airports 1997 [10]. It consists of 332 airports and 2126 edges among them. There is an edge linking a pair of airports iff in the year 1997 there was a flight company providing flights between those two airports. The size of a circle representing an airport in Fig. 2 is proportional to its degree— the number of airports linked to it. The airports with the largest degree are: Airport Chicago O’hare Intl Dallas/Fort Worth Intl The William B Hartsfield Atlanta Lambert-St Louis Intl Pittsburgh Intl

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For the overlap weight the edge cut at level 0.8 (a subnetwork of all edges with overlap weight at least 0.8) is presented in Fig. 3. It consists of two triangles, a path of length 2, and 17 separate edges. A tetrahedron (Kwigillingok, Kongiganak, Tuntutuliak, Bethel), see Fig. 4, gives the first triangle in Fig. 3—attached with the node Bethel to the rest of the network. From this example we see that in real-life networks edges with the largest overlap weight tend to be edges with relatively small degrees in their end nodes (o(e) = 1 implies deg(u) = deg(v) = t (e) + 1)—the overlap weight does not satisfy the condition ld3. Because of this the overlap weight is not very useful for data analytic tasks in searching for important elements of a given network. We would like to

Corrected Overlap Weight and Clustering Coefficient

7

emphasize here that there are many applications in which the overlap weight proves to be useful and appropriate; we question only its appropriateness for determining the most overlapped edges. We will try to improve the overlap weight definition to better suit the data analytic goals.

2.3 Corrected Overlap Weight We define a corrected overlap weight as o (e) =

t (e) . μ + M(e) − t (e)

By the definition of μ for every e ∈ E it holds t (e) ≤ μ. Since M(e) − t (e) ≥ 0 also μ + M(e) − t (e) ≥ μ and therefore ld2, 0 ≤ o (e) ≤ 1. o (e) = 0 exactly when t (e) = 0, and o (e) = 1 exactly when μ = M(e) = t (e). For ld3, the corresponding maximal edge neighbors subgraph contains T (0, μ, 0). The end nodes of the edge e are structurally equivalent. To show that ld1 also holds let G(1) (e) denote the edge neighbors subgraph of the edge e. Let f be an edge added to G(1) (e). We can assume that deg(u) ≥ deg(v), e = (u : v). Therefore M(e) = deg(u) − 1. We have to consider some cases: a. f ∈ E(G(1) (e)) : then G ∪ f = G and o (e, G ∪ f ) = o (e, G). b. f ∈ / E(G(1) (e)) : b1. f = (u : t) : then t ∈ N(v) \ T (e) \ e. It creates new triangle (u, v, t). We have t  (e) = t (e) + 1 and M  (e) = M(e) + 1. We get o (e, G ∪ f ) =

t (e) + 1 t  (e) = > o (e, G) μ + M  (e) − t  (e) μ + M(e) − t (e)

b2. f = (v : t) : then t ∈ N(u) \ T (e) \ e. It creates new triangle (u, v, t). We have t  (e) = t (e) + 1 and M  (e) = M(e). We get o (e, G ∪ f ) =

t (e) + 1 t  (e) =   μ + M (e) − t (e) μ + M(e) − t (e) − 1 >

t (e) + 1 > o (e, G) μ + M(e) − t (e)

b3. f = (t : w) and t, w ∈ N(u)∪N(v)\{u, v} : No new triangle on e is created. We have t  (e) = t (e) and M  (e) = M(e). Therefore o (e, G∪f ) = o (e, G). The corrected overlap weight o is a kind of local density measure, but it is primarily a substitutiability measure. To get a better local density measure we have to consider besides triangles also quadrilaterals (4-cycles).

8

V. Batagelj

2.4 US Airports 1997 Links with the Largest Corrected Overlap Weight For the US Airports 1997 network we get μ = 80. For the corrected overlap weight the edge cut at level 0.5 is presented in Fig. 5. Six links with the largest triangular weights are given in Table 1. In Fig. 6 all the neighbors of end nodes WB Hartsfield Atlanta and Charlotte/Douglas Intl of the link with the largest corrected overlap weight value are presented. They have 76 common (triangular) neighbors. The node WB Hartsfield Atlanta has 11 and the node Charlotte/Douglas Intl has 25 additional neighbors. Note (see Table 1) that there are some links with higher triangular weight, but also with a much higher number of additional neighbors—therefore with smaller corrected overlap weights.

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Fig. 5 US Airports 1997 links with the largest corrected overlap weight, cut at 0.5 Table 1 Largest triangular weights in US Airports 1997 network u Chicago O’hare Intl Chicago O’hare Intl Chicago O’hare Intl Chicago O’hare Intl The W B Hartsfield Atlanta The W B Hartsfield Atlanta

v Pittsburgh Intll Lambert-St Louis Intl Dallas/Fort Worth Intl The W B Hartsfield Atlanta Charlotte/Douglas Intl Dallas/Fort Worth Intl

t (e) 80 80 78 77 76 73

d(u) 139 139 118 101 101 101

d(v) 94 94 139 139 87 118

o (e) 0.57971 0.57971 0.55714 0.54610 0.73077 0.58871

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Fig. 6 US Airports links o (WB Hartsfield Atlanta, Charlotte/Douglas Intl) = 0.7308

2.5 Comparisons In Fig. 7 the set {(o(e), o (e)) : e ∈ E} is displayed for the US Airports 1997 network. For most edges it holds o (e) ≤ o(e). It is easy to see that o(e) < o (e) ⇔ μ < m(e). Edges with the overlap value o(e) > 0.8 have the corrected overlap weight o (e) < 0.2. In Fig. 8 the sets {(m(e), o(e)) : e ∈ E} and {(m(e), o (e)) : e ∈ E} are displayed for the US Airports 1997 network. With increasing of m(e) the corresponding overlap weight o(e) is decreasing, and the corresponding corrected overlap weight o (e) is also increasing. We can observe similar tendencies if we compare both weights with respect to the number of triangles t (e) (see Fig. 9).

3 Clustering Coefficient 3.1 Clustering Coefficient For a node u ∈ V in an undirected simple graph G = (V, E) its (local) clustering coefficient [9] is measuring a local density in the node u and is defined as a

10

V. Batagelj Overlap weights

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proportion of the number of existing edges between u’s neighbors to the number of all possible edges between u’s neighbors cc(u) =

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where E(u) = |E(N (u))|. If deg(u) ≤ 1, then cc(u) = 0.

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It is easy to see that E(u) =

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where S(u) = {e(u : v) : e ∈ E} is the star in node u. It holds 0 ≤ cc(u) ≤ 1; cc(u) = 1 exactly when E(N (u)) is isomorphic to Kdeg(u) —a complete graph on deg(u) nodes. Therefore it seems that the clustering coefficient could be used to identify nodes with the densest neighborhoods. The notion of clustering coefficient can be extended also to simple directed graphs (with loops).

3.2 US Airports with the Largest Clustering Coefficient Let us apply also the clustering coefficient to the US Airports 1997 network. In Table 2 airports with the clustering coefficient equal to 1 and the degree at least 4 are listed. There are 28 additional such airports with a degree 3 and 38 with a degree 2. Again we see that the clustering coefficient attains its largest value in nodes with a relatively small degree. The probability that we get a complete subgraph on N (u) is decreasing very fast with increasing of deg(u). The clustering coefficient does not satisfy the condition ld3.

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Table 2 US Airports 1997 with clustering coefficient = 1 n 1 2 3 4 5 6 7

deg 7 5 5 5 5 4 4

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n 8 9 10 11 12 13 14

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Airport Gunnison County Aspen-Pitkin Co/Sardy Field Hector Intll Burlington Regional Rafael Hernandez Wilkes-Barre/Scranton Intl Toledo Express

3.3 Corrected Clustering Coefficient To get a corrected version of the clustering coefficient we proposed in Pajek [11] to replace deg(u) in the denominator with  = maxv∈V deg(v). In this paper we propose another solution—we replace deg(u) − 1 with μ: cc (u) =

2 · E(u) , μ · deg(u)

deg(u) > 0.

If deg(u) = 0, then cc (u) = 0. Note that, if  > 0 then μ < . To verify the property ld1 we add to G(u) a new edge f with its end nodes in G(u). Then E  (u) = E(u) + 1 and deg (u) = deg(u). Therefore cc (u, G ∪ f ) =

2 · (E(u) + 1) 2 · E  (u) = > cc (u, G).  μ · deg (u) μ · deg(u)

To show the property ld2, 0 ≤ cc (u) ≤ 1, we have to consider two cases: a. deg(u) ≥ μ: then for v ∈ N(u) we have degN (u) (v) ≤ μ and therefore 2 · E(u) =

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v∈N (u)

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3.4 US Airports Nodes with the Largest Corrected Clustering Coefficient In Table 3 US Airports with the largest corrected clustering coefficient are listed. The largest value 0.3739 is attained for Cleveland-Hopkins Intl airport. In Fig. 10 the adjacency matrix of a subnetwork on its 45 neighbors is presented. The subnetwork is relatively complete. A small value of corrected clustering coefficient is due to relatively small deg = 45 with respect to μ = 80.

3.5 Comparisons In Fig. 11 the set {(cc(e), cc (e)) : e ∈ E} is displayed for the US Airports 1997 network. The correlation between both coefficients is very small. An important observation is that edges with the largest value of the clustering coefficient have relatively small values of the corrected clustering coefficient. We also see that the number of edges in a node’s neighborhood is almost functionally dependent on its degree. From Fig. 12 we see that the clustering coefficient is decreasing with the increasing degree. Nodes with a large degree have small values of the clustering coefficient. The values of corrected clustering coefficient are large for nodes of large degree. Table 3 US Airports 1997 with the largest corrected clustering coefficient

Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Value 0.3739 0.3700 0.3688 0.3595 0.3488 0.3457 0.3455 0.3429 0.3415 0.3405 0.3379 0.3359 0.3347 0.3335 0.3335

deg 45 50 56 42 61 70 67 53 47 42 56 46 62 41 50

Id Cleveland-Hopkins Intl General Edward Lawrence Logan Orlando Intl Tampa Intl Cincinnati/Northern Kentucky Intl Detroit Metropolitan Wayne County Newark Intl Baltimore-Washington Intl Miami Intl Washington National Nashville Intll John F Kennedy Intl Philadelphia Intl Indianapolis Intl La Guardia

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Chicago O’ Charlotte/ The Willia Newark Int Detroit Me Pittsburgh Philadelph BaltimoreOrlando In Cincinnati Nashville Lambert-St Dallas/For Tampa Intl Fort Laude Miami Intl La Guardia Washington General Mi Kansas Cit New Orlean Minneapoli Indianapol Houston In General Ed John F Ken Washington San Franci Phoenix Sk Mc Carran Stapleton Los Angele Seattle-Ta Bradley In Palm Beach Raleigh-Du Sarasota/B Southwest Greater Bu Theodore F Norfolk In Chicago Mi Louisville Yampa Vall Atlantic C

Chicago O’ Charlotte/ The Willia Newark Int Detroit Me Pittsburgh Philadelph BaltimoreOrlando In Cincinnati Nashville Lambert-St Dallas/For Tampa Intl Fort Laude Miami Intl La Guardia Washington General Mi Kansas Cit New Orlean Minneapoli Indianapol Houston In General Ed John F Ken Washington San Franci Phoenix Sk Mc Carran Stapleton Los Angele Seattle-Ta Bradley In Palm Beach Raleigh-Du Sarasota/B Southwest Greater Bu Theodore F Norfolk In Chicago Mi Louisville Yampa Vall Atlantic C

Fig. 10 Links among Cleveland-Hopkins Intl neighbors

4 Conclusions In this paper we showed that two network measures, the overlap weight and clustering coefficient, are not suitable for the data analytic task of determining important elements in a given network. We proposed corrected versions of these two measures that give expected results. Because μ ≤  we can replace in the corrected measures μ with . Its advantage is that it can be easier computed, but the corresponding corrected index is less “sensitive.”

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An interesting task for future research is a comparison of the proposed measures with measures from graph drawing [6–8]. Acknowledgements The computations were done combining Pajek [11] with short programs in Python and R [12]. This work is supported in part by the Slovenian Research Agency (research program P1-0294 and research projects J1-9187 and J7-8279) and by Russian Academic Excellence Project “5-100.” This paper is a detailed and extended version of the talk presented at the CMStatistics (ERCIM) 2015 Conference. The author’s attendance on the conference was partially supported by the COST Action IC1408—CRoNoS.

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References 1. Wasserman, S., Faust, K.: Social Network Analysis Methods and Applications. Structural Analysis in the Social Sciences. Cambridge University Press, Cambridge (1995) 2. Todeschini, R., Consonni, V.: Molecular Descriptors for Chemoinformatics, 2nd edn. WileyVCH, Weinheim (2009) 3. Onnela, J.P., Saramaki, J., Hyvonen, J., Szabo, G., Lazer, D., Kaski, K., Kertesz, J., Barabasi, A.L.: Structure and tie strengths in mobile communication networks. Proc. Natl. Acad. Sci. 104(18), 7332 (2007) 4. Holland, P.W., Leinhardt, S.: Transitivity in structural models of small groups. Comp. Group Stud. 2, 107–124 (1971) 5. Watts, D.J., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998) 6. Melançon, G., Sallaberry, A.: Edge metrics for visual graph analytics: a comparative study. In: 12th International Conference Information Visualisation, pp. 610–615 (2008) 7. Nocaj, A., Ortmann, M., Brandes, U.: Untangling the hairballs of multi-centered, small-world online social media networks. J. Graph Algorithms Appl. 19(2), 595–618 (2015) 8. Nocaj, A., Ortmann, M., Brandes, U.: Adaptive disentanglement based on local clustering in small-world network visualization. IEEE Trans. Vis. Comput. Graph. 22(6), 1662–1671 (2016) 9. Wikipedia: (2018). Clustering coefficient: https://en.wikipedia.org/wiki/Clustering_coefficient, Overlap coefficient: https://en.wikipedia.org/wiki/Overlap_coefficient, Hamming distance: https://en.wikipedia.org/wiki/Hamming_distance, Jaccard index: https://en.wikipedia.org/wiki/Jaccard_index 10. Batagelj, V., Mrvar, A.: Pajek data sets: US Airports network (2006). http://vlado.fmf.uni-lj.si/pub/networks/data/mix/USAir97.net 11. De Nooy, W., Mrvar, A., Batagelj, V.: Exploratory Social Network Analysis with Pajek; Revised and Expanded Edition for Updated Software. Structural Analysis in the Social Sciences. Cambridge University Press, Cambridge (2018) 12. Batagelj, V.: Corrected (2016). https://github.com/bavla/corrected

Bottom-Up Collegiality, Top-Down Collegiality, or Inside-Out Collegiality? Analyses of Multilevel Networks, Institutional Entrepreneurship and Laboratories for Social Change Emmanuel Lazega

Abstract This paper argues that the analysis of multilevel networks (AMN) is useful to understand politics, institutional entrepreneurship, and social change. AMN helps identify multilevel relational infrastructures (in particular multilevel social status) on which institutional entrepreneurship depends, especially in collegial oligarchies as laboratories for social change. In heavily bureaucratized societies, these laboratories take various forms such as bottom-up collegiality, top-down collegiality, and inside-out collegiality. We argue that, in an era of vital transitions, one of the main challenges for social network analyses is to use AMN to observe these collegial oligarchies and to model and understand social (in)capacities to build alternative multilevel relational infrastructures promoting social change. This challenge leads to another: that of understanding the conditions under which a form of collegiality is selected by contextualizing institutional entrepreneurship and its multilevel relational infrastructures. The paper theorizes organized mobility and relational turnover as important dimensions of this contextualization of institutionalization processes. Keywords Analysis of multilevel networks · Institutional entrepreneurship · Bottom-up collegiality · Top-down collegiality · Inside-out collegiality · Multilevel relational infrastructures · Organized mobility · Multispin

1 Introduction Historical transitions require new institutions. In this paper we first suggest that analyses of multilevel networks (AMN) provide a new understanding of institutional entrepreneurship. AMN offers models and methods for research designs based

E. Lazega () Sciences Po, CSO-CNRS, IUF, Paris, France e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. Ragozini, M. P. Vitale (eds.), Challenges in Social Network Research, Lecture Notes in Social Networks, https://doi.org/10.1007/978-3-030-31463-7_2

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on linked inter-individual and inter-organizational networks in which each of the superposed networks represents a level of collective agency. Individual members of one network belong to organizations of the other network through affiliation ties. This structural “linked design” [1] extends the sociological concept of duality [2] in which individuals and groups co-constitute each other. Generalizations of this formalism [3] craft a “formal theory of interpenetration” of levels. Articulation of distinct levels of action can be partly accounted for, beyond bipartite structures, with statistical analysis of such datasets [4, 5]. Resources exchanged at each level are of different types. Figure 1 represents a static multilevel network based on this design. We then show how AMN helps understand institutionalization processes in organized settings by identifying levels of collective agency as either bureaucratic or collegial, and key players or institutional entrepreneurs as active at two (or more) levels of agency simultaneously. They build and maintain multilevel relational infrastructures (MLRIs), in this case multilevel forms of social status. Such political processes and their negotiations are never routine, and therefore necessarily collegial. However, in a bureaucratized organizational society [6], collegiality is always combined with bureaucracy. We identify three multilevel combinations of bureaucracy and collegiality: bottom-up collegiality, top-down collegiality, and inside-out collegiality. Each characterizes a different kind of institutional entrepreneurship in a multilevel context. An example, that of the emergence of a new European institution (the Unified Patent Court), is used to illustrate top-down collegiality in institutional entrepreneurship. In this setting, a collegial oligarchy of judges with multilevel status, i.e., particularly active simultaneously at two levels of agency, i.e., a discrete cluster of “vertical linchpins” who are big fish in big ponds at the national and transnational levels, negotiates and imposes its conception of a new intellectual property regime for European economies. One of the scientific issues currently challenging social scientists studying social processes in dynamic multilevel networks is that institutional entrepreneurship has determinants and must be contextualized. Building and maintaining MLRIs is not a collective adventure that takes place in a vacuum. We argue that this contextualization must be approached with (and AMN models enriched with data on) at least two determinants of social processes in general: organized mobility of

Fig. 1 Big/little fish in big/small ponds: A multilevel network based on linked design. In these superposed networks, white nodes represent individuals, black nodes represent organizations, and ties between white and black nodes are affiliation ties of individuals to organizations. The size of the nodes represents their centrality scores in the network of their level of collective agency

Bottom-Up Collegiality, Top-Down Collegiality, or Inside-Out Collegiality. . .

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institutional entrepreneurs and relational turnover in their networks (OMRT) at each level of collective agency. A theory of the effect of OMRT on institutionalization requires intuitions on organized mobility of institutional entrepreneurs and relational turnover in their multilevel networks and inspiration for hypotheses are drawn from the metaphor of the “multispin.” In conclusion we argue that, by illuminating (understanding and explaining) the effect of OMRT and its dynamic invariants on the process of institution building, AMN should not only help model social (in)capacities to build new laboratories for social change but also help us understand how to keep this process accountable and democratic.

2 Politics, Analysis of Multilevel Networks, and Multilevel Relational Infrastructures Interdependencies between actors are too important in social life to be left unorganized, and actors and institutions struggle to organize them. Institutions are among the most venerable objects of study in the social sciences [7]. To simplify, institutions can commonly be defined as rules, norms, and beliefs that describe reality for actors, explaining what is and is not, what can be acted upon and what cannot, and how [8]. Contemporary thinking about the emergence of institutions is dominated in sociology by a variety of neo-institutional perspectives focusing on how norms promoted by institutional entrepreneurs elaborate taken for granted cultural categories, classifications, rules, and procedures that include beliefs and codes stabilizing action into routines [9]. Such a perspective has been shown to lack structure and agency [10–12]. Neo-structural sociology revisits this process by opening it to individual and collective agency, including work of organizing interdependencies. Social network analysis is then used, together with other methods, for tracking and understanding actors’ efforts to manage their interdependencies in contexts of cooperation and/or competition where interests diverge, conflicts flare up, constraining but often fragile and polynormative institutions are inherited from the past. As such it avoids reification of the notion of structure and helps in further developing a sociological theory of collective action and of the management of cooperation dilemmas [13–15]. Intentional, reflexive, and strategic behavior endogenizing the structure, not blind reproduction of underlying structure, are parts of the behavioral assumptions of this approach, including the use of organizations as “‘tools with a life of their own” and “dynamic conditioning fields” [16], i.e., as political communities in which new institutions are constructed. For social scientists, finding the links between structure, multilevel position, and collective agency in an organizational society is therefore still a complex task if it has to be carried out in a meaningful way, i.e., in a way that makes normative controversies, conflicts, and politics more intelligible. To do this, it is important

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to take into account the vertical complexities of the social world. This means differentiating between levels of collective agency and articulating their dynamics in measurements and models. Selznick’s “dynamic conditioning field” can thus be considered as a contextual effect and seen as a precursor to contemporary lines of research on multilevel stochastic actor-oriented models [17], multilevel exponential random graph models [5] and multilevel blockmodeling [18–20]. For sociologists accounting for these vertical complexities, rule-making is a complex multilevel political processes in which it is not always easy to identify who is responsible for the promotion of which rule, for example for successes or failures of a transnational regulatory regime. Observations of regulatory activities show that individuals with specific structural characteristics punch above their weight in terms of regulatory activity by precisely being active at two levels of management of interdependencies (advice and contract, for example) at the same time. Presence, participation, and decision-making activity at two or more levels simultaneously allow for cross-level influence at each level separately and, via such “vertical linchpins” [11], jointly. Whether or not such actors are accountable to others in similar ways at different levels, whether or not the rules that they promote are recognized as public goods, are important questions that theory and methodology should help address. Often rules are made discretely, and it takes very sharp stakeholders, experts, non-governmental organizations and journalists to evaluate them, with much regulatory inertia built into the system, much disagreement about whether or not a rule “works” in terms of protecting particular interests—especially the interests of the weakest parties. In sum the construction and/or maintenance of multilevel relational infrastructure become a step towards coordination within and between levels. Identifying some of the social realities for which multilevel networks are indicators leads to the notions of overlap and complementarity between levels. But it also shows that these levels co-constitute each other via the social construction of MLRIs. These MLRIs are vertical and horizontal differentiations between members (for example forms of multilevel social status of vertical linchpins and multilevel social niches) that are used to influence, from one level, events and processes at other levels. AMN is helpful in showing whether and how MLRI-based complex dynamics and coordination (between individuals, between organizations, and cross-level between individuals and organizations driving each other’s evolution) are the laboratories of institutionalization processes and social change in the organizational society. Dynamics of such multilevel systems of collective agency assume that the evolution of networks at one level of collective action is influenced by that of another level of collective action, and the other way around in recursive ways [1, 21–23]. Such dynamics can be considered to be the outcome of a meta-process bringing together both individuals and organizations, in which the evolution of one level explains in part (in causal terms) the evolution of the other. Level 1 relationships can emerge as a result of the emergence of level 2 relationships. Actors of level 1 may be able under certain circumstances to change the structure of level 2, especially by bringing MLRIs into the picture. MLRIs represent at the same time levers of

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institutional entrepreneurship and the locus of co-constitution between levels. This is where the two superposed systems of collective action co-evolve and adjust. As indicated, new families of models are needed to account for such dynamics. One family of models could be a multilevel extension of Snijders [24] model of dynamics of networks, using characteristics of level 2 network as set of exogenous factors in the evolution of level 1 network, and the other way around. At each step of the description of these dynamics at one focal level, information from other, lower or higher, levels must be integrated in the model. The co-evolution of both level networks is “added” to the co-evolution of behavior and relational choices. In terms of model specification, new “independent” variables from interorganizational networks operate at the inter-individual level, and vice versa. It is perhaps also worth extending Snijders’ multilevel version of the model of network dynamics, for example by introducing dual alters or induced potentials, i.e., extended opportunity structures [25], into this controlled formalism. A problem of “synchronization” between levels [26] also emerges. Synchronization is a task of scheduling and coordinating superimposed interpersonal and inter-organizational forms of collective agency, over time and at the cost of one of the levels. Social sciences are currently struggling to measure and model such synchronizations of time scales (short term, long term), especially in political processes where their manipulations can constitute an important competitive advantage. AMN and MLRIs, especially when they are dynamic, will help better understand politics and multilevel governance in the organizational society, in which superposed levels of collective agency operate, each following their own logic of coordination, while each level is also part of the context of the other levels. This is not trivial since these different logics can be, for example, bureaucratic vs collegial [27–29].

3 Bottom-Up Collegiality, Top-Down Collegiality, and Inside-Out Collegiality Indeed, organization sociology always starts from an analysis of work, understood in a broad sense as either routine or innovative. From this perspective, each level can be characterized as either predominantly bureaucratic or predominantly collegial [28]. In this dual logics approach, the bureaucratic model is meant to organize collective routine work, concentrate power, command and control unobtrusively at the top, and depersonalize interactions among members. The collegial model is meant to perform collective innovative work with uncertain, unpredictable output and help rival peers self-govern by trying to build agreements and by using private, personalized relational infrastructures to enforce these agreements. Because there are always non-routine tasks to be performed, including that of normative choices and institutional entrepreneurship, organizations are redefined as necessarily combining the two idealtypes for social discipline and productive efficiency, each with its

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formal and informal dimensions1 . Both models of organization are needed together in communities, workplaces, markets, and society, and their articulation is undertheorized and under-studied. These combinations of idealtypes must be reassessed in terms of their articulation in real life companies, associations, cooperatives, public authorities, etc. where they are perceived as legitimate or where their legitimacy is contested. A stratigraphic approach to organized settings [28] shows that “collegial pockets” as social niches capable of collective agency survive in dominant bureaucracies, although with very different and unequal levels of power in regulatory struggles: for example executive suites, professional departments, and workers’ trade unions. These collegial levels survive and operate in large bureaucratized and complex organizations when their members are able to come together and learn to defend their regulatory interests. Here AMN shows how the stratigraphic meeting of bureaucracy and collegiality in what we call bottom-up collegiality and top-down collegiality can use MLRIs to strengthen or undermine social participation in organized collective action. After two centuries of bureaucratization, collegiality as a generic form of organization is really a bottom-up type of collegiality [27] in which collegial pockets of peers—always characterized by oppositional solidarity challenging for incumbent rulers—try to build multilevel relational infrastructures and a presence in the levels of the bureaucracy in which decisive regulation takes place, including the executive level. In the predominantly bureaucratized contexts of contemporary societies, collegiality—where it still exists—is therefore more or less managerialized. Observations of how both models can complement and co-constitute each other in the sense that they drive each other’s evolution are provided in recent research. For example, focus on bureaucratic rotation of peers, a process that helps bureaucracies achieve stability from internal movement, provides a first empirical illustration of this dynamic combination. The case of a corporate law firm rotating associates among partners to achieve a balance of powers between rainmakers and schedulers struggling to regulate the organization illustrates this form of combination [10]. Because this combination of logics takes place in an already bureaucratized society, bottom-up collegiality, for example of professionals or trade union members, is often reshaped, and often neutralized, by the bureaucratic ruler, who transforms it into top-down collegiality [27]. The latter is a form of patronage characterized by collegial oligarchies composed by the ruler on a clientelistic basis. Top-down collegiality applies Selznick’s [16] cooptation bringing stakeholders into policymaking bodies, but forcing them to turn to this ruler (and to no one else) for help. MLRIs are thus often used by top-down collegiality and AMN is currently being used to provide efficient tools for studying them (for example [28, 29]).

1 To

avoid a frequent misunderstanding it is important to stress that collegiality is not the informal dimension of bureaucracy but the organizational idealtype orthogonal to that of bureaucracy. Both bureaucracy and collegiality have their own formal and informal dimensions, their own strengths and weaknesses or vicious cycles.

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In such dynamics of multilevel forms of organized collective agency, one particular and contemporary technological evolution deserves special attention for its social implications. One of the most phenomenal contemporary innovations is the digitalization of interactional and relational life with online social networks. In our view, these online networks boost the bureaucratic systematic capacity to monitor, reshape, and routinize collegial pockets, the very core makeup of collegiality: personalized relational activity and MLRIs. We call “inside-out collegiality” the combination of the two logics in which bureaucratic digital framing, parametrizing, monitoring and control of private personal relationships (made transparent to owners of the platform) shape collective agency in order to strip collegiality of its oppositional solidarity. In that sense, inside-out collegiality not only strengthens neo-liberal individualization and flexibilization of labor markets but threatens institutional entrepreneurship and the political process as defined above. The struggle and co-constitution between the two idealtypes thus takes a dramatic turn. Digitalization as contemporary bureaucratization turns the bottom-up collegial model “inside out,” deepening bureaucratization of collective action and society [30]. Freedoms and privacy, oppositional solidarities, and capacity to innovate are deeply threatened by what amounts to using organizations as tools for imposing new forms of collective responsibility and for further dividing societies between the many and the few [31]. Struggles to find new forms of collegiality in cooperatives, in the commons and in more distributed uses of platforms, such as new peer-to-peer innovations, resist such developments and would benefit from better knowledge of dynamics of multilevel networks in new forms of organized collective agency.

4 An Example of Top-Down Collegiality in Institutional Entrepreneurship An example can be provided in a study of the emergence of a new European intellectual property regime via the construction of a transnational court, the European Unified Patent Court (UPC) [32]. This court is considered by European industries with patents at the core of their business model as important to strengthening a contemporary European knowledge economy, including promotion and protection of innovation. The construction of this institution requires institutional entrepreneurship involving individuals (professionals), organizations, and governments. “Harmonization” of a variety of national legal frameworks has required MLRIs for coordination between networks of individuals, networks of organizations, and cross-level coordination between networks of individuals and organizations. Neither individuals, nor organizations, nor governments could access or mobilize, on their own and at the right time, all the resources that are needed to be efficient in this institutionalization process. Structuration at one level drove structuration at the other, often in conflicting and unequal ways. Time to adjust and adapt was available to some, but not to others in dynamic and multilevel political construction.

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Top-down collegiality accounts well for the construction of the UPC. With help from Brussels bureaucrats and from a professional association of corporate lawyers, a powerful, public-private European agency, the European Patent Office (EPO), sole regulator of intellectual property at the European level in the absence of a transnational court, a collegial oligarchy of national judges specialized in patents was selected as patent experts and assembled at the so-called Venice Forum, a private field-configuring event. Based on this top-down cooptation, a core group among these judges was then promoted as an ex ante leadership into a collegial oligarchy that was able to define the Rules of Procedure of the future UPC. They were punching above their weight in the regulatory process of harmonization of divergent national legal frameworks into a single body of rules under which the future institution would operate. The bureaucratic ruler in Brussels allied with EPO operated top-down through a form of patronage, selecting judges with strong multilevel status or promoting others to this status. This top down selection of a collegial oligarchy of ex ante leaders was instrumental for the development of the project, neutralizing in particular civil society actors opposed to the ways in which patents are used in contemporary capitalism, i.e. as financial instruments paradoxically undermining open science and increasingly innovation itself. Contemporary institutions are increasingly designed, operated and evaluated by such top down collegial oligarchies. One of the problems for such politics is precisely a problem of coordination of the regulatory processes that occur at one level with the same processes occurring at the other levels, i.e., “harmonization” of different time frames, sources of normativity and governance within and across levels. How this takes place is still not very well known in detail and can be investigated with AMN. Today, there are no tools for evaluating the vast dynamic and multilevel worldwide rule-making activity in any comprehensive way. New institutions arise when organized actors with sufficient resources see in them an opportunity to realize interests that they value highly [33, 34]. These “institutional entrepreneurs” struggle over which institutional arrangements to select for the collective. MLRIs and vertical linchpins driven by top-down collegiality in superposed levels of collective agency are thus key to policy- and rule-making is all domains of life: water management, food, health and safety, transportation, etc. An unknown number of discreet collegial oligarchies acquire the right kind of structural, cross-level position in such multilevel governance systems and create regulatory regimes that are not accountable to the public. In particular, institutional entrepreneurship requires a global vision of this multilevel system. Actors at different levels do not have the same resources and capacities to build this vision and to promote and protect their regulatory interests. In regulatory competition between strata of collective agency (local, national, international), the issue of how formal and informal knowledge networks and rulemaking behavior influence each other converge or diverge in terms of building institutions that will be considered to be legitimate, this issue is thus a crucial problem of dynamics of multilevel networks. The latter co-evolve with normative action taking place in several superposed political arenas, whether public, private (closed), or a mix of public/private, and are very complex to grasp. Usually,

Bottom-Up Collegiality, Top-Down Collegiality, or Inside-Out Collegiality. . .

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transnational private regulation that has been spreading globally pretends that it solves the problem of this competition between levels and stakeholders by providing flexible guidelines, a general normative baseline that is adaptable to local situations via subsidiarity, thus helping each level protect its regulatory interests as it sees fit. However these rhetorics are part of the process and need to be factored into the analyses as well, thus requiring dynamics of multilevel networks to combine structure, culture, and agency.

5 The Challenge of Contextualizing Multilevel Networks: Organized Mobility and Relational Turnover Building and maintaining MLRIs and institutions is not a collective adventure that takes place in a vacuum, but in Selznick’s [16] dynamic conditioning fields. To understand MLRIs and their role in synchronization of levels of collective action, it is useful to see them as determined in part by organizational mobility of members at each level and by subsequent relational turnover in their respective networks (OMRT). The word “organized” is used to qualify mobility because both social actors and the social system create paths and rules for movements and careers (for incoming, rotating, reshuffled, promoted, demoted, outgoing actors) that are not allowed to be random [35, 36]. Multilevel positioning can be complex because mobility in turn produces relational turnover for these members and this turnover is managed by the creation of the new relational infrastructures, for example specific forms of multilevel social status. Efforts to synchronize the temporalities of the levels create the energy for more intra- and inter-organizational mobility and controversies. Synchronization costs must then include efforts spent to position oneself in the dynamic conditioning fields at the different levels of social space so as to be able to build or maintain MLRIs. Incurring synchronization costs will be rewarding (in terms of managing constraints, learning, making one’s voice heard in controversies, and regulation) for some players; for others, who are unable to capitalize on social resources thanks to the maintenance of such multilevel relational infrastructures, they will amount to sunk costs. Actors can experience OMRT as new contextual constraints and opportunities, especially as possible emancipation from constraints imposed by prior affiliations, or as networks to nowhere, or as opportunities to introduce organizational change by bringing in new members. To some extent, institutional entrepreneurs attempt to use OMRT to reshape this multilevel structure—often with unexpected consequences. Such dynamics are not visible enough, for example, in current studies of social inequalities. A dynamic and multilevel network approach to social life changes the measurements of these socio-economic costs precisely by introducing more complex and systematic positioning, mobility, and relational turnover into the picture of management of inequalities.

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This assumes that some uses of MLRIs such as multilevel social niches and multilevel status (for example vertical linchpinship) are both building blocks for cross-level synchronization and instruments of restructuration attempts across levels. The connection between mobility and relational turnover is often explored in part and in depth in specific areas of social life. Often overlooked in the literature are the general effects of this systematic, recursive, and transformative link between the two realities (mobility across systems of places and relational capital) and its implications for social life. There are connections between these movements, as actors switch places in these circuits, and change—at least in part— their normative choices and respective sets of relationships, i.e., their respective relational capital. There is also an effect of the latter changes on the evolution of the system of places itself, an evolution that is only visible if places are not considered as purely contextual and exogenous, but as accumulated by actors—and thus as endogenous in the mechanisms under examination and models that account for them. Combining mobility in loops [35] and co-evolution of multilevel networks and behavior [37] helps make institutional entrepreneurship and OMRT structuration, with their multilevel dynamics and associated synchronization costs, measurable, and more generally redefine the social costs of living in an organizational and market society.

6 Multispin for Contextualizing Multilevel Networks Whether physical (for example through migration) or social or both, these articulated movements and changes represent important determinants of social structure, order, and inequalities in the organizational society. They are created by the social organization of these milieux and end up, under conditions that remain to be spelled out, restructuring these milieux, promoting some members in terms of ability to define new norms, and pushing others out of the regulatory process. This is where an overall theoretical link is needed between OMRT as forms of contextualization of networks, MLRIs, and organizational analysis of collective action. We propose, as an initial step, a guiding metaphor for this link in the picture of a multilevel spinning-top, or “multispin” (see Fig. 2). This metaphor is too rigid for many purposes, but helpful nevertheless as an initial heuristic for representation of the dynamic conditioning fields of institutionalization processes [36] because it is a dynamic structure combining several sub-processes in which movement creates stability, thereby promoting some actors and expelling others as in musical chairs. In our view, this image of a rotating three-level structure provides intuitions for contextualizing the emergence of institutions as a dynamic multilevel process. It helps explain how a small collegial oligarchy of networked institutional entrepreneurs with multiple and inconsistent forms of status [10] uses, in its lobbying activity, multilevel position in these networks and their dynamics. Stability from movement in the multispin helps institutional entrepreneurs acquire the staying capacity and subsequent influence that is needed to frame, build, and entrench new institutions.

Bottom-Up Collegiality, Top-Down Collegiality, or Inside-Out Collegiality. . .

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Fig. 2 Multilevel spinning-top with staircase in the shaft, or multispin, a metaphor for organized mobility and relational turnover. Design: Elie Partouche

In this metaphor, each level represents a network of collective action. In the emergence of institutions, the bottom level is composed of citizens, the second level of private organizations and public institutions, and the third level as governments, national and transnational. Affiliations as links between levels are not displayed in the picture for the sake of lisibility. Analytically speaking, agency starts at the level of individual networks. The evolution of these networks—each at its own level but influencing the evolution at the other levels through synchronization— is driven in part by controversies and mobility of actors moving into this system from the outside. The core set of individual institutional entrepreneurs with supercentral status moves up the shaft and acquires a competitive advantage in the joint regulatory process of institutionalization. Transferring synchronization costs is rewarding for these actors when they have a strong multilevel position because these costs are either shared or dumped on others, who can end up in the periphery or in limbo. For example, revolving doors from public responsibilities to private jobs and back to public positions help create this informal pecking order and concentrate power with help of conflicts of interests. In this example, it is not enough for institutional entrepreneurs to have an official mandate to build an institution. Superposition of these dynamic relational systems of collective action and coordinated activities between them must provide these super-central entrepreneurs with sufficient resources, staying capacity, stability, and legitimacy to drive the institution-building process over time, long enough for the institution to emerge and/or change. Multispin is a first metaphor meant to contextualize MLRIs and social processes that individual and collective actors navigate in Selznick’s dynamic conditioning (mine) fields. More generally, this metaphor accounts for the systematic rotation— such as job rotation—from one place to another in a system of places, a movement

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that creates relational turnover in members’ personal networks. Over time, this relational turnover tends to slow down because members manage turnover by turning to a small and stable set of authoritative contacts, for example super-central advisors, who can then be compared to members who climbed the stationary shaft of the multispin, a metaphor for social status as MLRI represented as a staircase. Indeed members can gain or lose multilevel status and vertical linchpinship, i.e., capacity to act at different levels simultaneously, just like stairs can lead up or down. They can rise upwards, usually to dominate, or sink downwards, usually to be pushed out of the regulatory process. They then gain or lose influence as institutional entrepreneurs because the tendency to turn to a small and stable set of authoritative contacts creates a central core at the next level higher up, ultimately becoming ratchets of social stratification [37]. In short, this metaphor brings together individual and collective actors, trajectories, relational turnover in actors’ networks, actors’ multilevel status measured by centrality in superposed, overlapping networks, decisions, and normative choices. This structure however can also lose its balance and the process fail, unless all these ingredients [38] are kept together by the energy coming from socially organized mobility, for example resilience from MLRIs. Multispin accounts for this institutionalization process in the empirical example presented above. Judges were brought to the Venice Forum on a top-down collegiality basis, then circulated across Europe to learn from each other and identify the ex ante leadership that was promoted to the collegial oligarchy with enough staying capacity, at least at two levels simultaneously, to become the permanent interlocutors of Brussels and EPO (the top level). Expert personnel was also circulated between corporate law firms, national ministries of justice, law schools in universities, courthouses, training facilities for future European judges, and industry associations, accounting for rotation at the medium inter-organizational level. Rotation at the level of governments was perhaps slower, less fluid than expected by the business communities bringing together large corporations, slowing down the process, and increasing synchronization costs for the levels below, to the point that the institutionalization process stalled, as if the multispin had stopped and fallen down.

7 Conclusion This paper argues that the analysis of multilevel networks is useful to understand politics, institutional entrepreneurship, and social change. Investigating these fundamental realities and phenomena requires combining inter-individual networks and inter-organizational networks of institutional entrepreneurship over time. AMN helps identify multilevel relational infrastructures (in particular multilevel social status) on which institutional entrepreneurship depends, especially in collegial oligarchies as laboratories for social change. In heavily bureaucratized societies, these laboratories can take various forms such as bottom-up collegiality, top-down

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collegiality, and inside-out collegiality. We argue that, in an era of vital transitions, one of the main challenges for social network analyses is to use AMN to observe these collegial oligarchies and to model and understand social (in)capacities to build alternative multilevel relational infrastructures promoting social change. This challenge leads to another: that of understanding the conditions under which a form of collegiality is selected by contextualizing institutional entrepreneurship and its multilevel relational infrastructures. The paper theorizes organized mobility and relational turnover as important dimensions of this contextualization of institutionalization processes. These analyses have the potential to play an important role in society, when faced with transitions-related challenges. In contemporary organizational societies, giant private companies create collegial oligarchies by using their privatized multilevel network data and instruments of inside-out collegiality for private institution building with questionable legitimacy: for example by reshaping entire cities with apparent democratization of new technologies of decentralization of services (blockchains); by developing private community self-organization with parametrized digital platforms for management of local resources, often competing with the public political architecture of these communities; by creating private currencies; by monopolizing relational data and building low quality social sciences (undermining high quality open science) for brute force social engineering. Studying Selznick’s “dynamic conditioning fields” as OMRT contextualizing forms of collegiality and institutional entrepreneurship might help understand these processes so as to keep multilevel political steering of future development accountable and democratic.

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11. Lazega, E.: Networks and institutionalization: a neo-structural approach. Connections. 37, 7–22 (2018) 12. Glückler, J., Suddaby, R., Lenz, R. (eds.): Knowledge and Institutions, Knowledge and Space Series, vol. 12. Springer, Dordrecht (2018) 13. Weber, M.: Economy and Society. University of California Press, Berkeley (1920/1978) 14. Olson, M.: The Logic of Collective Action. Harvard University Press, Cambridge (1965) 15. Wittek, R., Van de Bunt, G.G.: Post-bureaucratic governance, informal networks and oppositional solidarity in organizations. Neth. J. Soc. Sci. 40(3), 295–319 (2004) 16. Selznick, P.: TVA and the Grassroots: a Study in the Sociology of Formal Organization. University of California Press, Berkeley (1949) 17. Snijders, T.A.B.: The multiple flavours of multilevel issues for networks. In: Lazega, E., Snijders, T.A.B. (eds.) Multilevel Network Analysis for the Social Sciences; Theory, Methods and Applications, pp. 15–46. Springer (2016) 18. Žiberna, A.: Blockmodeling of multilevel networks. Soc. Networks. 39, 46–61 (2014) 19. Barbillon, P., Donnet, S., Lazega, E., Bar-Hen, A.: Stochastic block models for multiplex networks: an application to a multilevel network of researchers. J. R. Stat. Soc. A. Stat. Soc. 180(1), 295–314 (2017) 20. Bar-Hen, A., Barbillon, P., Donnet, S.: Block models for multipartite networks. Applications in ecology and ethnobiology. arXiv preprint arXiv:1807.10138 (2018) 21. Berends, H., Van Burg, E., van Raaij, E.M.: Contacts and contracts: cross-level network dynamics in the development of an aircraft material. Organ. Sci. 22, 940–960 (2011) 22. Berge, C.: Hypergraphs: Combinatorics of Finite Sets, vol. 45. Elsevier, Amsterdam (1984) 23. Grossetti, M.: L’espace à trois dimensions des phénomènes sociaux. Echelles d’action et d’analyse, SociologieS. http://sociologies.revues.org/index3466.html (2011) 24. Snijders, T.A.B.: Stochastic actor-oriented models for network change. J. Math. Sociol. 21, 149–172 (1996) 25. Lazega, E., Jourda, M.-T., Mounier, L.: Network lift from dual alters: extended opportunity structures from a multilevel and structural perspective. Eur. Sociol. Rev. 29, 1226–1238 (2013) 26. Lazega, E.: Synchronization costs in the organizational society: intermediary relational infrastructures in the dynamics of multilevel networks. In: Lazega, E., Snijders, T. (eds.) Multilevel Network Analysis: Theory, Methods and Applications. Springer, Dordrecht (2016) 27. Lazega, E., Wattebled, O.: Two definitions of collegiality and their inter-relation : The case of a Roman Catholic diocese. Sociol. Trav. 53(Supplement 1), e57–e77 (2011) 28. Lazega, E.: Bureaucracy, Collegiality and Social Change: Redefining Organization with Multilevel Relational Infrastructures. Edward Elgar, Cheltenham (2019) 29. Richard, C., Wang, P., Lazega, E.: A Multilevel Network Approach to Institutional Entrepreneurship: the Case of French Public-Private Partnerships (2019) 30. Lazega, E.: Networks and commons: bureaucracy, collegiality and organizational morphogenesis in the struggles to shape collective responsibility in new sharing institutions. In: Archer’s, M.S. (ed.) Morphogenesis and Human Flourishing, vol. V, pp. 211–237. Springer, Dordrecht (2017) 31. Lazega, E.: Swarm-teams with digital exoskeleton: On new military templates for the organizational society. In: Al-Amoudi, I., Lazega, E. (eds.) Post-Human Institutions and Organizations: Confronting the Matrix. Palgrave, Basingstoke (2019) 32. Lazega, E., Quintane, E., Casenaz, S.: Collegial oligarchy and networks of normative alignments in transnational institution building: the case of the European Unified Patent Court. Soc. Networks. 48, 10–22 (2016) 33. Eisenstadt, S.N.: Cultural orientations, institutional entrepreneurs, and social change: comparative analysis of traditional civilizations. Am. J. Sociol. 85(4), 840–869 (1980) 34. DiMaggio, P.: Interest and agency in institutional theory. In: Zucker, L. (ed.) Institutional Patterns and Organizations: Culture and Environments, pp. 3–21. Ballinger, Cambridge, MA (1988) 35. White, H.C.: Chains of Opportunity: System Models of Mobility in Organizations. Harvard University Press, Cambridge, MA (1970)

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Part I

Methods

Socio-Cultural Cognitive Mapping to Identify Communities and Latent Networks Iain Cruickshank and Kathleen M. Carley

Abstract Deriving networks and communities from individual and group attributes is an important task in understanding social groups and relations. In this work we propose a novel methodology to derive networks and communities from sociocultural data. Our methodology is based on socio-cultural cognitive mapping (SCM) and k-NN network modularity maximization (SCM + k-NN) that produces both a latent network and community assignments of entities based upon their socio-cultural and behavioral attributes. We apply this methodology to two realworld data sets and compare the community assignments by our methodology to those communities found by k-Means, Gaussian Mixture Models, and Affinity Propagation. We then analyze the latent networks that are created by SCM + k-NN to derive novel insight into the nature of the communities. The community assignments found by SCM + k-NN are comparable to those produced by current unsupervised machine learning techniques. Additionally, in contrast to current unsupervised machine learning techniques SCM + k-NN also produces a latent network that gives additional insight into community relationships. Keywords Social network analysis · Community detection · Clustering

1 Introduction An important task in social science research is to understand communities within a population and their relationship dynamics. Furthermore, these communities and relationships in the population are not known explicitly; we only have collected socio-cultural and behavioral attributes and must infer what communities exist and how they are related. In this work, we build upon previous work in socio-cultural cognitive maps (SCMs) to find these latent communities and network structures

I. Cruickshank () · K. M. Carley Center for Computational Analysis of Social and Organizational Systems (CASOS), Institute for Software Research, Carnegie Mellon University, Pittsburgh, PA, USA e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 G. Ragozini, M. P. Vitale (eds.), Challenges in Social Network Research, Lecture Notes in Social Networks, https://doi.org/10.1007/978-3-030-31463-7_3

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based upon recorded socio-cultural and behavioral attributes. Our approach provides a novel way of fusing together various socio-cultural and behavioral attributes into a network model that can be used with well-founded network analysis techniques to derive insight about real-world communities. In this work, we extend the SCM framework to find a latent graph and communities. The SCM framework finds the best fit coordinates in a N -dimensional space for a set of entities based upon an attenuated Minkowski distance applied to their observed attributes [1, 2]. With the coordinates produced by the SCM procedure, we can then measure the distance between points, such that those points which are closer are more similar in their socio-cultural attributes. Using that distance, we construct k-NN networks by considering the k nearest neighbors of each point. We then select those latent networks which produce the best modularity value for a modularity subgrouping routine [3]. The final result is both the latent network and community assignments for all of the socio-cultural data. Using this method, we apply it to two different real-world data sets and analyze the resultant latent communities and networks to demonstrate the utility of both the latent communities and latent networks in analysis of social science data.

2 Related Work The work presented in this paper falls within the general category of Latent Space Models. In general, latent space models typically work by placing entities in some kind of graph based upon the similarity (or distance) based on the observed attributes of the entities [4–6]. This is typically done by positioning nodes within an N dimensional space and then creating edges between those nodes that are closer together [6]. For nearly all latent space models, the underlying assumption of the models is that nodes within an N -dimensional space have ties between them based upon independent conditional probabilities [6, 7]. Regarding these models, there are generally two ways to estimate the probability distributions for edge formation: a distance model and a projection model [6]. In the distance model, nodes that are closer in a latent space tend to form links. In the projection model, nodes that have a narrower angle (as measured between two vectors extending from the origin to the points) between their spatial vectors tend to form links; a narrower angle implies that the points are closer in the latent space. These probabilistic models have been further extended to dynamic networks and multiplex or multidimensional networks [5, 6]. All of these models rely on the assumption that the formation of links between any given set of nodes in the latent space is statistically independent of any other link forming between any given set of nodes, which allows for fitting the models via Bayesian methods. When it comes to measuring communities or clusters within a network there are many measurements, such as conductance, expansion, internal density, and many others [8]. One of the most commonly used metrics is modularity [4, 9]. Roughly defined, modularity is the fraction of the links that fall within the defined communities of a network minus the expected fraction if links were distributed

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at random throughout the network. There have been many successful community detection algorithms for networks based upon the measure of modularity [10]. In this work, we will employ both the measure of modularity, as well as one of the more well-known and fast algorithms for modularity-based community detection, the Louvain Method, as part of the process of finding a network in the latent space. Another area of research that is closely related with our work is dimensionality reduction. Some classical examples of dimensionality reduction methods are Multidimensional Scaling (MDS) and Principal Component Analysis (PCA) [11]. Both of these techniques suffer from an assumption of linearity in their models. This shortcoming has been subsequently overcome by another common dimensionality reduction technique, t-Stochastic Neighbor Embeddings (t-SNE) [12]. Much like t-SNE, SCM embeds nodes in a particular space of a different number of dimensions than the original data (usually, fewer dimensions), which enables visualization and clustering [2]. As noted in previous work on SCMs, there are some distinct differences between SCM and more standard dimensionality reduction procedures [2]. In particular, SCM does not rely on specification of a distance metric for the latent space, but rather determines one through its optimization procedure. This flexibility allows SCM to vary the amount of impact the position of one node has on position of other nodes when placing nodes into a latent space. Additionally, unlike t-SNE which uses statistics for continuous or binary data to evaluate fit, SCM is custom suited to handle categorical data, which occurs in sociological and behavioral studies.

3 Method To begin, we will briefly describe the SCM process, but also refer the reader to [1] for the precise details of the formulation of an SCM process and [2] for an implementation of an SCM process. The first step in the SCM process will be to convert the data matrix into a frequency matrix. As an example, a data matrix could be a set of actors with their associated attributes, and at each attribute, an actor can have a series of categorical responses. It should be noted that all attributes are considered equal; there is no distinction between behavioral and socio-cultural attributes. This matrix is then converted to a frequency matrix where the entries are the counts of the shared levels of every attribute between each of the actors. So, the frequency matrix is of size actor-by-actor, and has entries that are the count of the number of same attribute values for all attributes of the data matrix between each actor and every other actor. Next, we then find the positions in a latent space for each of the data points such that the function determining the distances produces fitted frequencies that are as near as possible to the observed frequencies from the previous step. The fitted frequencies for each data point are calculated by F (i, j ) = Ri × Cj × 2−dij a

(1)

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where Ri is the row factor term, Cj is the column factor term, and 2−dij is the row and column correlation factor, where a is an attenuation term. a controls how much impact the interaction between two entities has on their co-occurrence frequency. A smaller a results in the distance affecting the frequency count to a smaller extent as the distance between any given data points increases. When a = 2 the interaction between row and column variables is modeled as a Gaussian Distribution. The distance metric, dij is defined as the Minkowski distance: a

dij =

 ndim 

1/M |xik − yj k |

M

(2)

dim=k

where M is the power setting of the metric that determines the space that the points are mapped into (i.e., M = 2 produces an Euclidean distance and space). So, the SCM process then proceeds as an optimization problem where the points of each actor are inferred in a latent space with the target function as the χ 2 value between the fitted frequencies, F (i, j ) and the observed frequencies, for user supplied attenuation, power, and number of dimensions. The overall process of the SCM can be summarized as follows (Fig. 1 further illustrates the SCM procedure): 1. Select a set of power and attenuation settings and number of dimensions for the SCM procedure. Typical selections are a = 0.7, 1, 2, 3 and M = 0.7, 1, 2, 3, and two dimensions for the inferred points of the agents. 2. For each combination of the parameters of a and M, do the following: – Place each data point in a latent space where the number of dimensions of the latent space is specified by the user, and calculate the frequencies of this placement using Eqs. 2 and 1.

Fig. 1 Flowchart of the SCM process. User inputs are the number of dimensions, power, and attenuation, and outputs are the latent points of the nodes and the distances between the nodes in the latent space

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– Evaluate the current placement using the result of the last step and observed frequencies using a χ 2 value. – Continue to adjust the placements of the points to achieve a better χ 2 value. 3. Return the coordinates of those placements of the points that have the best χ 2 value. Once we have obtained the pairwise distances between each of the actors, we can then use k-NN network modularity maximization to find the latent network and subgroup assignments of the actors. We will leave the technical details of the algorithm to [3] and [13], but will give a brief description of the algorithm, with our changes to it. k-NN network modularity maximization takes an affinity or distance matrix and creates a network where each node connects to its k nearest neighbors. Then, this network is clustered using the Louvain method of modularity maximization [14]. In clustering step, we differ from the original algorithm proposed in [3], as we use a faster method of modularity maximization of unimodal networks, the Louvain Method, as opposed to the author’s QCut algorithm. For our methodology, we are using an asymmetric k-NN network, where two nodes are connected if one or both of the nodes feature the other node in their k nearest neighbors [13]. More precisely, for each point xi , let Nk (xi ) be the k nearest neighbors of xi , then an asymmetric k-NN network has links between two points xi and xj if xi ∈ Nk (xj ) OR xj ∈ Nk (xi ). The pseudocode of our implementation of the k-NN modularity maximization procedure is detailed in Algorithm 1. Algorithm 1 k-NN modularity maximization procedure input: Distance or Affinity Matrix, S, (n x n) for i = 1 : log2 n do k ← 2i Gk ← kN N (S, k) C(Gk ) ← Louvain(Gk ) Grk ← randomize(Gk ) C(Grk ) ← Louvain(Grk ) Modularityk ← Modularity(C(Gk )) − Modularity(C(Grk )) end for k ∗ ← argmaxk Modularityk G∗ ← kN N (s, k ∗ ) C(G∗ ) ← Louvain(G∗ ) return C(G∗ ), G∗

With these latent networks and subgroups, we can then address our initial questions regarding possible groups and network structures in a community based on measured attributes of the community. To summarize, the SCM plus k-NN modularity maximization procedure works as follows: 1. Select a set of power and attenuation settings and number of dimensions for the SCM procedure. Typical selections are a = 0.7, 1, 2, 3 and M = 0.7, 1, 2, 3, and two dimensions for the inferred points of the agents.

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2. Run the SCM procedure for each of the attenuation and power settings combinations. Select the output from those settings that produce the minimal χ 2 value. Find the distance between each point and every other point based upon the coordinates found by the SCM procedure, and store these in a distance matrix. 3. Using the distance matrix, perform k-NN network modularity maximization to find the optimal subgroups and latent network. 4. Analyze the subgroups and latent networks using standard network analysis techniques to answer research questions.

4 Empirical Results To evaluate the SCM plus k-NN network modularity maximization procedure (SCM + k-NN), we will consider two real-world data sets and compare the subgroups obtained to those obtained by other unsupervised techniques. Since we are interested in methods that also determine the number of subgroups without a priori input, we will consider only other unsupervised machine learning techniques that do not require specification of the number of subgroups. Namely, we will use k-Means, where k is determined by Bayesian Information Criteria (BIC) [15], a Gaussian Mixture Model where the number of Gaussians is determined BIC [16], and affinity propagation [17]. For each of the experiments, we used the program Organizational Risk Analyzer (ORA) version 3.0.9 to perform the SCM with settings of a = 0.7, 1, 2, 3 and M = 0.7, 1, 2, 3 with two dimensions for the inferred points in the latent space [18]. All visualizations and standard network analysis techniques were also performed in ORA. For the other three methods, similarity was determined using Euclidean distance. Since these data sets do not have a ground truth subgrouping, we will analyze the subgroups for qualitative meaning and compare different subgrouping results using the Adjusted Rand Index (ARI) [19]. ARI is a measure of the similarity of two clusterings; it gives an idea of how often two different methods will assign the same labels to the same sets of data. We will also use Silhouette score to measure the performance of each of the clustering techniques on each of the data sets [20]. The two data sets are as follows: – Hatfield McCoy Data Set: Data set is a set of actors with their associated attributes from the famous historical feud between the Hatfield and McCoy families from 1863 to 1891 [2, 21]. The data set was compiled from various historical records and includes individuals attributes from the time of the feud (1863–1891). It consists of 66 individuals with 9 attributes that are all binary valued (e.g., member of Hatfield family, Female, etc.) – 8th Convocation of the Ukrainian Parliament: Data set consists of the set of the parliamentarians of the 8th (current) convocation of the Ukrainian Parliament with some of their publicly available attributes and their votes on the bills that were proposed by the Ukrainian president [22]. The data set has 522

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parliamentarians with their votes on 62 different presidential bills. Vote responses can take one of five values: yea, nay, abstain, absent, did not vote (some members of the parliament were not members of the parliament for all of the convocations’ votes). The data set also contains 10 categorical and real-valued attributes, such as age, former professions, etc.

4.1 Hatfield–McCoy Case Study Beginning with the Hatfield McCoy data set, the optimal SCM parameters were a = 2, M = 0.7, and a goodness-of-fit of χ 2 = 32.36. Table 1 summarizes the summary statistics of the different subgroups identified in the data by the different methodologies: Affinity propagation and SCM + k-NN both seem to have produced very similar subgroups, with a larger number of subgroups relative to both GMM and k-Means. Furthermore, both affinity propagation and SCM + k-NN produced higher values of the silhouette which indicates less overlap between subgroups than those subgroups found by GMM or k-Means. Looking at the membership of the different subgroups, k-Means seems to have almost entirely split upon whether the person was a Hatfield or a McCoy; the two subgroups are the two families. The subgroups produced by the other three methods have much more nuance. For example, SCM + k-NN has separate subgroups for the “Devil Anse Kids” and the “Randolph Kids” which are both subsets of the Hatfield’s and the McCoy’s, respectively. We next analyze the subgroups found by the various methods by ARI, as depicted in Fig. 2. As noticed in the summary statistics the subgroup assignments of individuals people are highest between SCM + k-NN and affinity propagation. k-Means had subgroup assignments least like any other subgrouping method. Therefore, it would seem SCM + k-NN produced meaningful and more nuanced subgroups comparable with affinity propagation. One distinct advantage of SCM + k-NN modularity, however, is that we also obtain a latent network, which is visualized in Fig. 3. In the resulting network, two of the most interesting groups of individuals, the “Devil Anse Kids” and the “Randolph Kids,” are placed into completely separate components. Additionally, running standard unimodal network centrality measures produces some interesting results. Ephraim Hatfield and America McCoy, both of

Table 1 Summary statistics of subgroups found in Hatfield McCoy data set

Subgrouping method SCM+k-NN GMM with BIC k-Means with BIC Affinity propagation

Number of subgroups 11 4 2 11

Avg. subgroup size 6.0 16.5 33.0 6.0

Std. subgroup size 1.89 6.61 4.24 3.16

Range of subgroup sizes [3,10] [12,26] [30,36] [3,11]

Silhouette 0.411 0.319 0.378 0.691

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Fig. 2 ARI plots of subgroups for (a) Hatfield McCoy and (b) Ukrainian Parliament

Fig. 3 Latent network of the Hatfield McCoy data set. Coloring represents subgroup assignments. The low, uniform values of degree for the nodes indicate that actors are spread out in their sociocultural attributes than sharing many of the same socio-cultural attributes. Areas of high sociocultural similarity would more likely result in a more hub-and-spoke style of network

whom were intermarried to the members of the other family, ranked highest in betweenness centrality, which likely reflects the bridging roles they played by being intermarried. Floyd Hatfield, Henry Hatfield, and Mary Hatfield have the highest degree centrality values in the latent graph, which indicates the Hatfields have more central and shared socio-cultural attributes relative to the entire population than the McCoys. Furthermore, nearly all of the Hatfield members are within the same component, whereas the McCoy members are split more evenly across multiple components of the latent graph. This property of the latent graph may indicate that the Hatfields possess more socio-cultural similarity than the McCoys, which could have impacted the historical course of events of the Hatfield McCoy feud. Additionally, while there are two main components in the network, these components are

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not densely interconnected, possibly indicating that the socio-cultural latent space is more spread out and homogenous and there are no particularly dense regions of socio-cultural attribute values. If each actor were allowed to increase its number of links, it is likely the modularity of the overall graph and subgroups would decrease.

4.2 Ukrainian Parliament Case Study Turning now to the second data set of the 8th Convocation of the Ukrainian Parliament, the optimal SCM parameters were a = 2, M = 2.0, and a goodnessof-fit of χ 2 = 88,896.69. Table 2 reports the summary statistics of the different subgroups identified in the data. As with the previous data set, we find first that the grouping statistics between affinity propagation and SCM + k-NN are more alike and the grouping statistics between k-Means and GMM are more alike. These differences are particularly pronounced in the sizes and number of the subgroups found in the data. This difference in size and number of groups is likely also a contributor to the relative differences in the silhouette score between the two sets of methods. The generally lower values of the silhouette score for this data set likely indicate that it is a difficult data set to subgroup, as there is a lot of overlap between the attributes and behavior of parliamentarians, and so finding a smaller number of subgroups has likely led to more overlap between the subgroups. One major difference with the SCM + kNN procedure on this data set is that SCM + k-NN produced more subgroups than any other method, especially when compared to GMM and k-Means. This would seem to suggest that the method is more sensitive to subtler relationships between different combinations of socio-cultural and behavioral attributes. As with the previous data set, we will also look at how similar subgroup assignments are between methods using the ARI scores, which are depicted in Fig. 2. Additionally, since a parliamentarian’s stated faction is often used to define the factions within the parliament, we have also included the stated formal faction assignments as a subgrouping procedure for comparison. The ARI plot clearly demonstrates that GMM and k-Means produce the exact same subgroup assignments on this data set. This is good evidence that there is likely a good

Table 2 Summary statistics of subgroups found in Ukrainian Parliament data set

Subgrouping method SCM+k-NN Gaussian Mixture Model with BIC k-Means with BIC Affinity Propagation

Number of subgroups 40 2

Avg. subgroup size 13.05 261

Std. subgroup size 8.09 124.45

Range of subgroup sizes [3,28] [173,349]

Silhouette 0.18 0.37

2 29

261 18.0

124.45 10.43

[173,349] [7,55]

0.37 0.10

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split between two major subgroups within the parliament, but beyond those two major splits, finding subgroups becomes increasingly difficult. Also, no subgrouping compared well with the formal parties that the parliamentarians affiliate with. Thus, it would seem that while party is an important component in the subgroups of a political body like the Ukrainian Parliament, it is certainly not the only sociocultural or behavioral component that influences actual subgroup formation. Lastly, while no subgrouping method is very similar in subgroup assignments, outside of GMM and k-Means, affinity propagation is the most similar to the other methods, on average. That would suggest that there is almost a hierarchy of subgrouping methods on this data set when it comes to the granularity of subgroups found. The methods of GMM and k-Means find very course, large subgroups and the methods of SCM + k-NN find very fine and nuanced subgroups. Affinity propagation find subgroups of granularity somewhere in between the other four methods. As with previous data set, using SCM + k-NN also produces a latent graph which is visualized in Fig. 4. Most of the components in this network tend to form around

Fig. 4 Latent network of the Ukrainian Parliament data set. Coloring represents subgroup assignments, and key personalities are labeled in the network [22]

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voting habits. The component containing Boyko Yuriy Anatoliyovych is routinely absent for or against most presidential bills, which would indicate that particular component is an opposition faction. And, while Boyko Yuriy Anatoliyovych is the leader of the “Opposition Bloc” political party, the members of his component are not all members of the “Opposition Bloc.” That particular component also contains members of the “Revival” party and unaffiliated members of the parliament. Additionally, the component containing Turchynov Alexander Valentinovich, who is a member of the “People’s Front” also did not participate in many of the presidential proposed votes. This component which is very inhomogenous in terms of party affiliation may represent an unofficial opposition. Analyzing the network with standard social network analysis measures we observe some interesting results from the latent network. Dzhemilev Mustafa, Lopushansky Andriy Yaroslavovich, Bandurov Volodymyr Volodymyrovych, and Gerega Alexander Vladimirovich all score highly in betweenness centrality. Two of the four members are members of the Peter Porchenko Bloc and a different two are on the Committee on Energy Complex, Nuclear Policy and Nuclear Safety. All four of them seem to have varied voting records between them. As such, these individuals may represent important moderate elements of the parliament as they do not particularly ally with any faction but have behavior and attributes characteristic of many members of the parliament. Alekseev Sergey Olegovich, Artyushenko Igor Andreevich, Bublyk Yuri Vasilievich, and Boyko Olena Petrovna all score highly in degree centrality. Three of the four all come from the “Peter Porchenko Bloc.” This would seem to indicate that the “Peter Porchenko Bloc” may wield a good deal of influence in the parliament due to the socio-central members of its party.

5 Discussion The SCM + k-NN method finds a best-fit latent space for entities based on their socio-cultural attributes and behaviors and then finds a latent network and subgroup assignments by iteratively building k-NN graphs and evaluating the modularity of the subgroups and graphs. This method tended to produce smaller, more finely grained subgroups than any of the other methods we compared it to in this paper. In particular, for the Ukrainian Parliament data set, the SCM + k-NN found many more, and more nuanced subgroups than either GMM or k-Means when k was determined by BIC. This would suggest that the SCM + k-NN method would complement social science research in understanding populations from socio-cultural and behavioral data by finding nuanced subgroups and relationships between those subgroups. However, it also suggests that the method may be sensitive to noise present in the data, which is certainly a limitation in practical research. The sensitivity to noise in the data likely also affected the relatively high number of components; 8 for Hatfield–McCoy and 26 for the Ukrainian Parliament. If the method were less sensitive, there would be fewer components and the latent network would likely better illustrate community dynamics. Additionally, no distinction was made

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between socio-cultural attributes and behavioral attributes when performing SCM. This is a current limitation of the method and a direction for future research. Our method also provides a novel method of constructing latent space networks. As opposed to traditional latent space models that are fit to data using maximum likelihood estimators, our proposed method fits a network to the data based on best fit distances and network characteristics (i.e., modularity). The links produced by this method tend to reflect neighborhoods of similarity of socio-cultural variables. Since the links only form to those entities who are nearby in the latent socio-cultural space, and only form when it supports local neighborhood development, measure by modularity, the latent network will emphasize neighborhoods of socio-cultural and behavioral similarity. Since socio-cultural similarity can be an important driver of link formation in social networks [9], these networks may then better represent a possible unobserved social network for a given population.

6 Conclusion and Future Work Overall, our proposed method of SCM + k-NN produces meaningful subgroups on the two real-world data sets we analyzed. Usage on two real-world data sets demonstrated the utility of the subgroups, and latent network, that the method found. SCM + k-NN’s performance in clustering was also different than existing unsupervised machine learning techniques. In particular, SCM + k-NN matched the performance of affinity propagation far more than it matched GMMs or kMeans. The end result of our proposed procedure is the possible relations and sub-communities that could exist in a population based upon just their shared behavior and socio-cultural attributes, which is useful for analysis in social science research. One particular avenue of future research that we did not address in this paper is to use other unsupervised techniques with SCM. While we did show the utility of the k-NN network modularity maximization procedure to operate on SCMs with the SCM + k-NN method, it remains to be tried whether constructing the graph and subgrouping the graph based on other measure than modularity would work better with SCMs to create more connected latent graphs. Additionally, when using the SCM procedure there was no distinction between behavioral and sociocultural variables; all variable contributed equally to the frequency counts. It would be interesting to adapt the SCM procedure in the future to somehow differentiate between these classes of variables. Finally, the data sets we considered for the case studies in this paper had a relatively modest number of attributes; 9 for the Hatfield– McCoy and 62 for the Ukrainian Parliament. It would be interesting to investigate how well the SCM procedure performs when we are able to obtain hundreds or thousands of attributes for each individual. Acknowledgements This material is based upon work supported by the National Science Foundation Graduate Research Fellowship (DGE 1745016), Department of Defense Minerva Initiative

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(N00014-15-1-2797), and Office of Naval Research Multidisciplinary University Research Initiative (N00014-17-1-2675). Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation, Department of Defense, or the Office of Naval Research.

References 1. Levine, J.H., Carley, K.M.: SCM system. Technical Report. CMU-ISR-16-108, Institute for Software Research, Carnegie Mellon University (June 2016). http://reports-archive.adm.cs. cmu.edu/anon/isr2016/CMU-ISR-16-108.pdf 2. Morgan, G.P., Levine, J., Carley, K.M.: Socio-cultural cognitive mapping. In: Lee, D., Lin, Y.R., Osgood, N., Thomson, R. (eds.) Social, Cultural, and Behavioral Modeling, pp. 71–76. Springer International Publishing, Cham (2017) 3. Ruan, J.: A fully automated method for discovering community structures in high dimensional data. In: 2009 Ninth IEEE International Conference on Data Mining, pp. 968–973 (Dec 2009). https://doi.org/10.1109/ICDM.2009.141 4. Barabsi, A.L.: Network Science. Cambridge University Press, Cambridge (2015). http:// networksciencebook.com/ 5. Snijders, T.A.B., Lazega, E.: Multilevel Network Analysis for the Social Sciences. Springer, Cham (2016) 6. Kim, B., Lee, K., Xue, L., Niu, X.: A review of dynamic network models with latent variables (2018). arXiv. https://arxiv.org/pdf/1711.10421.pdf 7. Hoff, P.D., Raftery, A.E., Handcock, M.S.: Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97(460), 1090–1098 (2002). https://doi.org/10.1198/016214502388618906 8. Leskovec, J., Lang, K.J., Mahoney, M.W.: Empirical comparison of algorithms for network community detection. CoRR abs/1004.3539 (2010). http://arxiv.org/abs/1004.3539 9. Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010) 10. Schaub, M.T., Delvenne, J., Rosvall, M., Lambiotte, R.: The many facets of community detection in complex networks. CoRR abs/1611.07769 (2016). http://arxiv.org/abs/1611.07769 11. Jolliffe, I.: Principal Component Analysis. American Cancer Society (2005). https://doi.org/ 10.1002/0470013192.bsa501, https://onlinelibrary.wiley.com/doi/abs/10.1002/0470013192. bsa501 12. van der Maaten, L., Hinton, G.: Visualizing high-dimensional data using t-SNE. J. Mach. Learn. Res. 9, 2579–2605 (2008) 13. Maier, M., Hein, M., von Luxburg, U.: Optimal construction of k-nearest neighbor graphs for identifying noisy clusters. Theor. Comput. Sci. 410(19), 1749–1764 (2009). https://arxiv.org/ abs/0912.3408 14. Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp. 10 (2008). http://arxiv.org/abs/0803.0476 15. Pelleg, D., Moore, A.: X-means: extending k-means with efficient estimation of the number of clusters. In: In Proceedings of the 17th International Conference on Machine Learning, pp. 727–734. Morgan Kaufmann, San Francisco (2000) 16. Marin, J.M., Mengersen, K.L., Robert, C.: Bayesian modelling and inference on mixtures of distributions. In: Dey, D., Rao, C. (eds.) Handbook of Statistics, vol. 25. Elsevier, Amsterdam (2005). https://eprints.qut.edu.au/901/ 17. Frey, B.J., Dueck, D.: Clustering by passing messages between data points. Science 315(5814), 972–976 (2007). https://doi.org/10.1126/science.1136800, http://science. sciencemag.org/content/315/5814/972 18. Center for Computational Analysis of Social and Organizational Systems: ORA (June 2018). http://www.casos.cs.cmu.edu/projects/ora/

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Bootstrapping the Gini Index of the Network Degree: An Application for Italian Corporate Governance Carlo Drago and Roberto Ricciuti

Abstract We propose a new approach based on bootstrapping to compare complex networks. This is an important task when we wish to compare the effect of a (policy) shock on the structure of a network. The bootstrap test compares two values of the Gini index, and the test is performed on the difference between them. The application is based on the interlocking directorship network. At the director level, Italian corporate governance is characterized by the widespread occurrence of interlocking directorates. Article 36 of Law 214/2011 prohibited interlocking directorates in the financial sector. We compare the interlocking directorship networks in 2009 (before the reform) with 2012 (after the reform) and find evidence of an asymmetric effect of the reform on the network centrality of the different companies but no significant effects on Gini indices.

1 Introduction During the past decades, many scholars have come up with theories to explain the presence of interlocking directorates (board members that simultaneously sit on more than one board: for a review of the literature see [1, 2]). From an economic standpoint, interlocking directorates are important because they can increase collusion among different companies whose directors sit on their respective boards, reducing consumer welfare. The effectiveness of “busy” board members sitting on several boards may also diminish, with less ability to check the chief executive officer’s decisions, exposing companies to high risks [3]. Network analysis has been applied several times to the analysis of Italian corporate governance and ownership [4–9].

C. Drago Niccolò Cusano University, Rome, Italy e-mail: [email protected] R. Ricciuti () University of Verona, Verona, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. Ragozini, M. P. Vitale (eds.), Challenges in Social Network Research, Lecture Notes in Social Networks, https://doi.org/10.1007/978-3-030-31463-7_4

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The chapter is organized as follows. In Sect. 2 we introduce our approach with simulations performed in Sect. 3. Section 4 presents our application, analyzing the effect of a reform banning interlocking directorates in banks and insurance companies in Italy, to see whether this led to significant changes in the structure of the network after the reform. Section 5 concludes.

2 The Approach Given a network G = (V, E) where V indicates the vertices of the network and E the edges, for each node we calculate the degree [10]: CD (v) = deg(v)

(1)

The degree is the number of nodes that are their neighbors. In this way, each node is characterized by its local measure of centrality [11]. Then for each network, we compute the Gini inequality index [12]:    n 1 j =1 (n + 1 − i) xj n Gk = n+1−2 n−1 j =1 yj

(2)

where n is the number of nodes, x is the degree that characterizes the jth node. We have perfect equality if the Gini index is equal to 0, which means that all the degrees show the same value. For each network k we obtain different values for the Gini index. The bootstrap test compares two Gini indices, G1 and G2 , and tests the statistical significance of the difference D = G1 − G2 [13, 14]. It is important to note that we can bootstrap the distribution of G1 or G2 similarly to bootstrapping the distribution of D [13]. Following [13], it is important to note that the bootstrap distribution Fˆ (D) allows us to obtain the values for the hypothesis testing on D. With the bootstrap hypothesis testing method [15], we draw samples from the original data with replacement from data to obtain the bootstrap sample. Then we approximate the distribution of D by the bootstrap distribution of D∗ . Finally, from the sampling distribution of D by obtaining their bootstrap estimate hypothesis testing [16, 17] can be carried out. Finally, the results are used to test whether the change in inequality in degree can be considered statistically significant. The code is written in Stata [18] and in R [19, 20].

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3 Simulations To try out the methodology, we simulated some networks on which we performed the algorithm to observe the changes that occur when impacted by structural changes.1 We consider different networks and scenarios to test whether significant changes over time can be identified. The simulation is performed by randomly generating six networks and then executing the bootstrap test. In each test, we use two network topologies with the same number of nodes. In the first set, we start by considering two extreme cases: the first with a highly centralized network structure (a star) versus a structure evolved in an Erdös Renyi model [21] and the second a typically centralized (the Barabasi Albert model [22]) and a non-centralized network (the Erdös Renyi model). We then provide further experiments by increasing the number of nodes/edges to construct more complex structures that may react differently to a shock. In this way, a more challenging environment is created for the null hypothesis. The second batch of simulations is based on the evolution of a network over time in which the deletion of some edges is simulated, followed by the addition of other edges in the second round. More precisely, we start from a Barabasi Albert Model and then randomly delete 4 edges in the first round and add 12 edges in the second. The initial network for both simulations is represented in Fig. 1. In the second experiment, we consider a higher additional number of edges by randomly removing and adding edges of the first network evolution simulation, i.e., 4 edges are deleted and 20 added. We consider all possible connections on the nodes. The adjacency matrix is related to all possible connections which can occur in the network. Addition can occur randomly at each theoretically plausible connection and if the edge already exists, it is kept, otherwise, a new link is added. Therefore, the edge is added only where the connection is non-existent. The final networks are shown in Fig. 2 (simulation 1) and Fig. 3 for simulation 2. It is interesting to note that very different network structures can be quickly obtained with the addition or deletion of the single edges. We simulate the network and compute the degree for each node, then calculate the Gini index for both networks. Finally, we consider the third batch of simulations. We simulate the destruction (first case) and creation (second case) of a random number of edges between 100 and 1. We plot the initial networks (Figs. 4 and 5) and their final state after the destruction (Fig. 6) and the construction (Fig. 7) of new edges.

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the simulations we have considered very general networks in which there are no effects caused by preferential attachment, triadic closure, clustering tendencies, constraints on the degree distribution due to transaction costs, which may characterize economic networks. The different additions/destructions of the links occur randomly. In the future, we will consider more complex cases that consider these features. These more complex structures may lead to complex reactions to the addition/destruction of the different nodes. In particular, the process of addition/destruction may be quicker or slower with the implication that the test may be more or less likely to reject the null hypothesis.

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Coef. Std. err. Z P > |z| Centralized structure (star model) versus Erdös-Renyi model −0.19101 0.04908 −3.89 0.000 Model with similar structures (Barabasi-Albert models) −0.02894 0.08562 −0.34 0.735 Network evolution (1) −0.06268 0.05898 1.06 0.288 Network evolution (2) −0.11300 0.04575 −2.47 0.014 Network edge deletion 0.07047 0.10186 0.69 0.489 Network edge creation −0.23972 0.07925 −3.02 0.002 Number of observations: 400, replications 2000

In our simulations (Table 1), the null hypothesis of a similar structure can be rejected in the first and simplest case but not in the second. In other words, in the first case we found a significant effect on the Gini index of the degree of the network, but not in the second case. In the two more complex simulations, the null hypothesis in Network evolution 1 cannot be rejected, whereas it may be slightly rejected in Network evolution 2. Finally, we are unable to reject the null hypothesis for network edge deletion, but we can reject the null for network edge creation.

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4 Application The application is based on the interlocking directorates network. Italian corporate governance features a high concentration of ownership and the presence of controlenhancing mechanisms that are conducive to controlling shareholders’ dominance and exploitation of minorities. At the director level, corporate governance is characterized by the widespread recourse to interlocking directorates (directors sitting on more than one board at the same time, hereafter referred to as ID). Through cross-ownerships, circular ownerships, and interlocking directorates, the Italian system has been characterized by pyramidal groups headed by a small number of families that permanently control the firms. A few reforms have been implemented over the last 15 years to open up the market for corporate control, reduce the scope for collusion, and to protect minorities from exploitation by controlling shareholders able to extract private benefits at the expense of the minority. The latest addition to this wave of reforms was a new measure introduced in 2011, article 36 banning interlocking, part of the Save Italy Decree which started life as Legal Decree 201/2011, published in Official Gazette on December 6, 2011. This decree was converted into law 214 with amendments in 2011 and published in the Official Gazette on December 27, 2011. Under article 36 paragraph 2b the requirement of 120 days to comply with the law ran from December 27, 2011. Therefore, the director of a bank or insurance company with incompatible appointments was required to choose one of the two (or more) positions by April 27, 2012, and failing this, would lose all the positions. The effects of the Law were in place when the data for our study were collected (December 31, 2012); hence, it is legitimate to compare 2012 with 2009 to see if the provision was effective in reducing ID in the financial sector. The reform aimed to break the ties between the sectors, increasing competition between financial companies. If this were true, we would observe a sparser network after the reform, with a significantly lower concentration in the Gini Index and more communities. Data were collected from listed companies in light of the Board of Directors for each firm on 31/12. Only the management board is considered for the few companies with the two-tier system. We used publicly available data from Consob (the Italian stock market regulator) and to collect the network data, we considered individual names and the related company and then created the twoway matrix, from which we were able to perform the one mode projection to obtain the adjacency matrices both for the network of directors and for the network of companies. A weighting represents the number of directors shared by connected companies.2 Nodes represent companies. The isolates are companies that do not share any director with other companies.

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with the projected two-mode would make it possible to analyze the inequality in degree both on the company and the director side.

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Fig. 8 The network in 2009

Fig. 9 The network in 2012

Figures 8 and 9 show the networks in the 2 years of interest. Visually comparing the structures of the networks, we find a group of nodes connected with each other on a first component and several isolates. Table 2 reports the descriptive statistics for the two networks. We then perform a community detection analysis to decompose the networks in communities. We use the walktrap community detection approach [23] because it

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Nodes Edges Density Islands\clusters Global cluster coefficient Diameter Betweenness (mean) Degree (mean)

Network 2009 278 576 0.01496 66 0.252179 10 225.4018 4.143885

Network 2012 251 387 0.012335 80 0.260765 15 162.8853 3.083665

Fig. 10 Network in 2009: community structure

can detect and separate the different communities in this context. Walktrap is an appealing approach in our context because the random walks can be related to the dynamics of the information in the network. The first community detection relating to the year 2009 (Fig. 10) shows 105 communities. Communities vary in size from 49 for the largest group to 1 for the isolates. The result shows that there are different groups of nodes strongly connected to each other. The communities tend to connect weakly compared to dense intracommunity networks, and the network is utterly dissimilar to a random graph. Hence the expectation that the nodes have a different number of connections and different centrality in the same network. This result is particularly important because it confirms the need to consider the Gini index analysis to investigate the structure of the distribution of the degree over time.

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Fig. 11 Network in 2012: community structure

Regarding the community structure of the network in 2012 (Fig. 11), the number of communities increases to 111. There is greater fragmentation of the communities, with the largest community including 32 nodes against 1 for the isolates. The heterogeneity in the distribution of the links between the nodes tends to decrease from 2009 to 2012 regarding the degree, for the first network it is in a range of 1 to 34, whereas in 2012 the range is from 1 to 21. In addition, the median and the mean are higher in 2009 (4.00 and 5.144, respectively) than in 2012 (3.00 and 4.084, respectively). Before the reform, the average value of the degree for each node is higher, possibly related to a different structure in the two networks. Finally, the variance of the degree for 2009 and 2012 moves from 21.71 to 13.43. Given the observed reduction in the heterogeneity of the degree, this may have an impact on its distribution, and this calls for an analysis of the Gini index of the degree in the 2 years. The coefficient observed is 0.02 and the bootstrap standard error is 0.02575, with 2000 replications in the bootstrap process. Therefore, we detect no statistical significance, since we cannot reject the null hypothesis at 5%. In this case, we have included only the nodes which show a positive degree because we considered only nodes with at least a linkage, neglecting isolates. If we consider all the nodes, isolates, and non-isolates and repeat the analysis, we obtain a Gini index for the year 2009 of 0.55 and for 2012 of 0.60. These results are interesting because the deletion of some links due to the reforms has created an increase in the Gini index. The computed coefficient is 0.04151 with the bootstrap standard error equal to 0.02881. In this case too, we cannot reject the null hypothesis at the 5% significance level. We also plot the results obtained for the Lorenz Curves (Fig. 12) comparing the

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Fig. 12 Lorenz Curves (black line for 2009, dotted line for 2012)

results for the year 2009 (black line) and for 2012 (dotted line). Overall, no strong effect of the reform on the network can be found (as expected; see [6]), whereas an asymmetric effect on the distribution of the edges (by considering both connected nodes and isolated) is evident. The asymmetric effect is due to the presence of some nodes that became more central in a local sense, because of edge deletion. Interestingly, this potentially shows some unintended effects of the Law: by deleting some edges, some nodes became even more important than before (see [24] showing this effect in the period 1998–2006 for S&P MIB financial companies). This result is similar to [25], where community detection techniques for the analysis of the networks in 2009 and 2012 ascertained the effect of the reform on the network of Italian directorates. They find that, although the number of interlocking directorates decreases in 2012, the reduction takes place mainly at the periphery of the network. The result is due to the fact the creation/deletion process fails to activate the “structural change threshold.”

5 Conclusions This chapter presents a new method for the detection of statistically significant changes in the network structure by a bootstrap test of the degree. The approach appears to be very useful in analyzing the impact of exogenous shocks on a network and in detecting impacts on the network structure. In our case, the shock was a statutory amendment aimed at cutting the links between banking and insurance companies in terms of directors sitting on more than one board. The methodology can be extended to other node-level statistics, such as closeness, betweenness, and so on. It is also possible to think about the centrality not only considering single nodes but also groups of nodes (entire communities). The advantage of this approach

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is the ability to observe the relative significance of shocks which can occur in a network system over time and space. The approach is also promising because it can be applied to other network structural measures such as betweenness. Possible problems can be identified in the fact that bootstrapping is not robust to outliers. Our results show the limited effects of this legislative measure on the network of companies, possibly suggesting that a more far-reaching intervention was needed to achieve the desired outcome. Another possible reason could be that interlocking directorates are a symptom of cross-shareholding, and therefore regulation aimed at breaking these networks should address the former rather than the latter. Future research should try to understand the ultimate causes of the intertwining of listed Italian companies. Acknowledgments We would like to thank Paolo Santella and Antonio Balzanella for useful discussions and two anonymous reviewers for constructive comments. Any remaining errors are ours alone. A previous version of the paper was presented at the SIS 2015 Conference.

References 1. Drago, C., Millo, F., Ricciuti, R., Santella, P.: Corporate governance reforms, interlocking directorship and company performance in Italy. Int. Rev. Law Econ. 41, 38–49 (2015) 2. Enriques, L.: Corporate governance reforms in Italy: what has been done and what is left to do. Eur. Bus. Organ. Law Rev. 10, 477–513 (2009) 3. Fich, E., Shivdasani, A.: Are busy boards effective monitors? J. Financ. 61, 689–724 (2006) 4. Battiston, S.: Inner structure of capital control networks. Physica A. 338(1–2), 107–112 (2004) 5. Battiston, S., Catanzaro, M.: Statistical properties of corporate board and director networks. Eur. Phys. J. B. 38(2), 345–352 (2004) 6. Caldarelli, G., Catanzaro, M.: The corporate boards networks. Physica A. 338(1–2), 98–106 (2004) 7. Bellenzier, L., Grassi, R.: Interlocking directorates in Italy: persistent links in network dynamics. J. Econ. Interac. Coord. 9, 183–202 (2014) 8. Piccardi, C., Calatroni, L., Bertoni, F.: Communities in Italian corporate networks. Physica A. 389(22), 5247–5258 (2010) 9. Vasta, M., Drago, C., Ricciuti, R., Rinaldi, A.: Reassessing the bank–industry relationship in Italy, 1913–1936: a counterfactual analysis. Cliometrica. 11(2), 183–216 (2017) 10. Badham, J.M.: Commentary: measuring the shape of degree distributions. Netw. Sci. 1, 213– 225 (2012) 11. Wasserman, S.: Social Network Analysis: Methods and Applications, vol. 8. Cambridge University Press, New York (1994) 12. Forcina, A., Giorgi, G.M.: Early Gini’s contributions to inequality measurement and statistical inference. Electron. J. Hist. Probab. Stat. 1, 31 (2005) 13. Abdon M.: Bootstrapping Gini. http://statadaily.ikonomiya.com/2011/05/23/ bootstrappinggini/ 14. Mills, J., Zandvakili, S.: Statistical inference via bootstrapping for measures of inequality. J. Appl. Econ. 12(2), 133–150 (1997) 15. Efron, B.: The Jackknife, the Bootstrap and Other Resampling Plans, vol. 38. Society for Industrial and Applied Mathematics, Philadelphia (1982) 16. Efron, B., Tibshirani, R.J.: An Introduction to the Bootstrap. CRC Press, Boca Raton (1994)

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17. Biewen, M.: Bootstrap inference for inequality, mobility and poverty measurement. J. Econ. 108, 317–342 (2002) 18. Jenkins, S.P.: INEQDECO: Stata module to calculate inequality indices with decomposition by subgroup. Statistical Software Components S366002, Boston College Department of Economics (1999), revised 22 Jan 2015 19. Csardi, G., Nepusz, T.: The igraph software package for complex network research. InterJournal Complex Syst. 1695, 1 (2006). http://igraph.org 20. Iacus, S. M., Masarotto, G.: labstatR: Libreria del Laboratorio di Statistica con R. R package version 1.0.7. http://CRAN.R-project.org/package=labstatR (2012) 21. Erdös, P., Renyi, A.: On random graphs. Publ. Math. 6, 290–297 (1959) 22. Barabasi, A.L., Albert, R.: Emergence of scaling in random networks. Science. 286, 509–512 (1999) 23. Reiczigel, J., Zakari’as, I., R’ozsa, L.: A bootstrap test of stochastic equality of two populations. Am. Stat. 59(2), 156–161 (2005) 24. Santella, P., Drago, C., Polo, A.: The Italian chamber of lords sits on listed company boards: an empirical analysis of Italian listed company boards from 1998 to 2006, MPRA Paper No. 2265 (2009) 25. Drago, C., Ricciuti, R.: Communities detection as a tool to assess a reform in the Italian interlocking directorship network. Physica A. 466, 91–104 (2017)

Association Rules and Network Analysis for Exploring Comorbidity Patterns in Health Systems Giuseppe Giordano, Mario De Santis, Sergio Pagano, Giancarlo Ragozini, Maria Prosperina Vitale, and Pierpaolo Cavallo

Abstract The presence of patients affected by different diseases at the same time is becoming a major health and societal issue. In clinical literature, this phenomenon is known as comorbidity, and it can be studied from the administrative databases of general practitioners’ prescriptions based on diagnoses. In this contribution, we propose a two-step strategy for analyzing comorbidity patterns. In the first step, we investigate the prescription data with association rules extracted by a twomode network (or bipartite graph) to find frequent itemsets that can be used to assist physicians in making diagnoses. In the second step, we derive a one-mode network of the diseases codes with association rules, and we perform the k-core partitioning algorithm to identify the most relevant and connected parts in the network corresponding to the most related pathologies. Keywords Comorbidity pattern · Health data · Network data

1 Introduction Today, the presence of patients affected by different diseases at the same time is becoming a major health and societal issue, because the presence of chronic or multi-morbid diseases is growing. In clinical literature, this phenomenon is known

G. Giordano () · M. P. Vitale Department of Political and Social Studies, University of Salerno, Fisciano (SA), Italy e-mail: [email protected] M. De Santis Cooperativa Medi Service, Salerno, Italy S. Pagano · P. Cavallo Department of Physics E.R. Caianiello, University of Salerno, Fisciano (SA), Italy G. Ragozini Department of Political Science, University of Naples Federico II, Naples, Italy © Springer Nature Switzerland AG 2020 G. Ragozini, M. P. Vitale (eds.), Challenges in Social Network Research, Lecture Notes in Social Networks, https://doi.org/10.1007/978-3-030-31463-7_5

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as comorbidity, that is, “any distinct additional entity that has existed or may occur during the clinical course of a patient who has the index disease under study” [12]. A possibility of studying comorbidity using available datasets resides mainly in administrative databases of general practitioners’ (GPs’) prescriptions. A recent literature review [28] showed that comorbidity can be studied from these databases based on either diagnoses, using the International Classification of Diseases (ICD) codes or pharmacy data. The comorbidity data are then described according to the co-occurrence of diagnostics in a single prescription as relationships between diagnoses. In this scenario, the present contribution aimed at exploiting the GP administrative databases for health-care systems analysis, to mine the complexity of this information by using statistical techniques to extract the relevant features. A two-step strategy for analyzing comorbidity patterns is proposed based on association rules and social network analysis. In the first step, by considering the data structure as a transactional dataset, the prescription data can be investigated with the association rules extracted by the two-mode network to find frequent itemsets that can be used to assist physicians in making diagnoses. In the second step, the association rules can be represented as a one-mode network in which the diagnoses are the node, and the links are given by the rules. Thus, the one-mode network of the disease codes is analyzed by partitioning algorithms, to identify the most relevant and connected parts in the network structure corresponding to the most related pathologies. This paper is organized as follows: Section 2 presents the theoretical framework. Section 3 introduces the data, and how they are used to define the comorbidity networks. Section 4 describes the strategy for the analysis. Section 5 reports the main findings, while Sect. 6 briefly discusses future lines of research.

2 Theoretical Background Literature has differentiated two core concepts, comorbidity and multimorbidity. The latter is defined as the coexistence of two or more long-term conditions in an individual [19] not biologically or functionally linked, while the former concept, comorbidity, defines the coexistence of conditions that are actually linked [25], either biologically or functionally. If we define two different conditions, one called the “index condition” (IC) [5] and the other the “comorbid condition” (CC), in a state of comorbidity there will be one IC with one or more CCs. In contrast, as multimorbidity is usually defined as the co-occurrence of multiple chronic or acute diseases and medical conditions within one person without any reference to an index condition, in such a situation there will be a number of ICs and no CCs, but each IC could theoretically be the “seed” of a series of CCs.

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Comorbidity, moreover, has been intended [20] in a positive or negative sense, namely syntropy and dystrophy [21]. The former is the mutual disposition, or the attraction of two or more diseases in the same individual, while the latter indicates those pathologies that are rarely found in the same patient at the same time. Thus, a given IC can attract one or more CCs to which it is syntropic, but at the same time, repulsing one or more different CCs to which the IC is dystropic. In the case of multimorbidity, two or more ICs not syntropic to each other can be syntropic or dystropic to a given number of CCs. Time span and sequence [25] should be considered: a different length of cooccurrence and/or the appearance of the same two conditions in different sequences, e.g., A as the IC and B as the CC or vice versa, may imply different outcomes although the comorbidity conditions are the same. Simply considering this taxonomy, the intrinsically complex nature of comorbidity can be easily understood. If one considers the limitations of models and instruments used to represent this phenomenon, the following question becomes crucial [9]: “What do we observe when two disorders covary: a genuine phenomenon that is independent of our diagnostic criteria, measurement scales, and measurement models, or (in part) an artifact of the structure of these criteria and models?” In this sense, it has been suggested that research could usefully focus on behavioral medicine and secondary analyses of available datasets [24], and a standardized classification, such as the ICD, can be used as the basis to do that, taking into account the emerging construct of patient complexity. On this basis, the morbidity and comorbidity burden is considered to be influenced by several factors, namely health-related characteristics and socioeconomic, cultural, environmental, and behavioral characteristics, but there is a lack of agreement [5] on how to understand the complex interdependent relationships between diseases, due to the large number of variables (many of which are latent), the lack of accuracy in measurements, and the technological limitations in generating data. For example, acute cardiovascular hospitalization Medicare claims data [8] have been successfully considered for understanding and measuring either the presence of comorbid conditions and of function-related indicators, such as depression, walking impairment. In addition, data from a network of GPs have been used to study disease clustering for a group of prevalent diseases, such as diabetes and osteoarthritis, showing a significant increment of the prevalence of comorbidity with respect to the patients’ age. In Italy, GPs can prescribe drugs, laboratory tests, imaging tests, specialist referrals, and hospitalization. For each type of prescription, a specific comorbidity network can be extracted by considering the co-occurrence of diagnostics in a single prescription as relationships between diagnoses. Thus, different patterns could emerge according to demographic or epidemiological factors. To the best of our knowledge, nobody has investigated these network aspects of comorbidity using an administrative GP database.

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3 The Data In the present contribution, the electronic health recordings (EHRs) of the prescriptions made by a group of 10 GPs are considered in the analysis, corresponding to a total number of 14,958 patients, and covering a time interval of 12 years, from 2002 to 2013. All the data used in the study were provided in an anonymous form, either for patients and GPs, according to the Italian law on privacy and guidelines of the Declaration of Helsinki. The study was retrospective observational, with anonymous data analyzed in aggregate form. The relevant ethics committee granted approval (Comitato Etico Campania Sud, document number 59, released on 2016-06-08). As the outcome of a patient’s visit to a GP there is generally a prescription, containing a series of items of various types: drugs, laboratory tests, imaging tests, etc. The GP administrative prescription data used to provide, for each patient visit, the following information: – – – –

patient ID, a unique random number assigned to the patient; demographic data, age and sex; prescription date; prescription type, drug, laboratory test, imaging, specialist referral, and hospitalization; – prescription code, a specific code for each prescription type; and – associated ICD diagnostic code, the pathology connected to the specific prescription. The total number of analyzed prescriptions was 1,728,736, and their categorization by type is reported in Table 1. Each item present in a GP prescription has, according to Italian National Health System rules, an associated possible disease encoded using the ICD, Ninth Revision, Clinical Modification (ICD-9-CM) [27]. The ICD is the standard diagnostic tool for epidemiology, health management, and clinical purposes, including the analysis of the general health situation of population groups. The simple count of codes is used to monitor the incidence and prevalence of diseases and other health problems, providing a picture of the general health situation of countries and populations. It has the general form xxx.yy, where xxx is the general disease, and yy is a specific occurrence, for example, 250 is the code for diabetes, and 250.91 is the code for

Table 1 Number of prescriptions by type

Prescription type Drug Lab Others Total

Number of prescriptions 897,329 647,023 184,384 1,728,736

Mean number of prescriptions per patient 59.99 43.26 12.32 115.57

Mean number of prescriptions per patient in 1 year 5.00 3.61 1.02 9.63

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Table 2 ICD-9-CM groups From 1 140 240 280 290 320 360 390 460 520 580 630 680 710 740 760 780 800 E00 V00

To 139 239 279 289 319 359 389 459 519 579 629 679 709 739 759 779 799 999 E99 V99

Acronym INFE NEOP META BLD MENT NERV SENS CIRC RESP DIGE GEN PREG SKIN MUSC CONG NEWB ILL INJ EXT SUPP

Description Infectious and parasitic diseases Neoplasms Endocrine, nutritional and metabolic diseases, and immunity disorders Diseases of the blood and blood forming organs Mental disorders Diseases of the nervous system Diseases of the sense organs Diseases of the circulatory system Diseases of the respiratory system Diseases of the digestive system Diseases of the genitourinary system Complications of pregnancy, childbirth, and the puerperium Diseases of the skin and subcutaneous tissue Diseases of the musculoskeletal system and connective tissue Congenital anomalies Certain conditions originating in the perinatal period Symptoms, signs, and ill-defined conditions Injury and poisoning External causes of injury Supplemental classification

diabetes type 1 (juvenile) with unspecified complication, not stated as uncontrolled. Only the general form, the first three digits, of ICD codes was used, grouped following the structure of the ICD-9-CM classification, to obtain 20 groups, each representing a specific epidemiological area. The 20 groups are listed in Table 2.

4 The Analytic Strategy The comorbidity data are described according to the co-occurrence of diagnostics in a single prescription as relationships between diagnoses. At this end, a twomode network is derived by considering the ICD-9-CM diagnostic codes and the prescriptions as two disjoint sets of nodes. The diagnoses are linked if corresponding codes appear in prescriptions for the same patient on the same day. The sex and age of patients, and the type and the time of the prescriptions, can be considered attributes of a given prescription. Specifically, in the present paper, we analyze men and women separately, as sex is a common risk factor in determining different pathology patterns. In this preliminary analysis, we consider only adult patients older than 35 years, without further differentiation.

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More formally, the two-mode network can be represented as a bipartite graph B consisting of the two sets of relationally connected nodes, and can be represented by a triple B (D, P, A), with D denoting the set of ICD-9-CM codes, P the set of prescriptions, and A ⊆ D × P the set of ties. As we are interested in the comorbidity network, the two-mode network can be projected in a one-mode network diagnoses × diagnoses [11]. In such a case, two diagnoses are connected if they are present in the same prescription for the same patient, and the weight of the link is given by the number of patients in which this relationship is present. In Fig. 1 two different one-mode networks for women

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Fig. 1 One-network of pathologies obtained from administrative databases of General Practitioners prescriptions for women (a) and men (b)

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and men, respectively, are plotted. The two networks are very dense, and present the well-known hairball effect. For this reason, it seems necessary to extract and explore the most interesting parts of the comorbidity network by considering a twostep strategy of analysis, as described below.

4.1 Association Rules Mining the Prescriptions Dataset As the first step to disentangle the complexity in the comorbidity data, we extracted the association rules [2] from the two-mode network data. This technique is strictly related to the aim of finding frequent itemsets in a large dataset, and commonly applied in transactional data for marketing strategies. Association rule mining from transaction data [2], implication rules for market basket data [4], and recommendation technology [1] are labels used in the literature to point out the search, visualization, and analysis of frequent itemsets in a large dataset. Transaction data arise in many business firms, in economic and commercial activity in daily operations. Association rules mining aims at discovering important relations between itemsets in the form of frequent items. For instance, an itemset can be the set of all products sold at a discount; on the other side, the set of buyers is the active set that operates by selecting items from the products set (the passive set). Such selections can be easily arranged in a table T where each row holds the elements of the active set (customers) and the columns are the passive set items (products). Each entry of this matrix indicates with 1 the selected product, and 0 otherwise. In this case, the interest could be in discovering whether subsets of products are bought together in a large percentage of cases, and derive suitable recommendations for similar buyers. This example drawn from a basket analysis situation can be generalized to different applicative fields. Applying association rules in medical diagnosis can be used for assisting physicians to make diagnoses. Even if reliable diagnostic rules are difficult, and may result in hypotheses with unsatisfactory prediction, too unreliable for critical medical applications [14, 23] proposed a technique based on relational association rules and supervised learning methods. It helps to identify the probability of illness in a certain disease. In a study of protein structures, Gupta et al. [15] deciphered the nature of associations between different amino acids that are present in a protein. Such association rules are desirable for enhancing our understanding of the protein composition, and hold the potential to give clues regarding the global interactions among some sets of amino acids occurring in proteins. Formally, let D = {d1 , d2 , . . . , di , . . . , dM } be the set of all items (products or, in our case, either diagnoses or diseases); let P = {p1 , p2 , . . . , pj , . . . , pN } be the set of all transactions (prescriptions) where each transaction pj is a subset of items chosen from D, (pj ⊂ D), so that a prescription that had more diagnoses can be considered a transaction. A collection of items is termed an itemset. A prescription pj contains an itemset X if X is a subset of pj . The number of transactions that contain the particular itemset X defines the support count of X.

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An association rule is an implication expression of the form X → Y , where X and Y are disjoint itemsets of D: X ∩ Y = ∅, X, Y ∈ D. The strength of an association rule can be measured in terms of its support, confidence, and lift. – The support of itemset X ⊂ D is the fraction of prescriptions in P, of which X is a subset. It represents the prevalence of diagnoses (simple or multiple) in the universe of prescriptions P, and can be stated as a relative frequency. A threshold s can be fixed so that when the support of a set of diagnoses is at least equal to the fixed value s, then the itemset is said to be frequent. The itemsets that exceed the minimum support s are said to be frequent patterns. These patterns can provide useful information about associations in prescriptions and diagnosis, in the case of simple itemset, that is, (X ≡ di ). – The confidence of a rule C(d1 → d2 ) is the proportion of prescriptions with diagnosis d1 , in which diagnosis d2 also appears. It is a conditional frequency, and can be stated in terms of the ratio between the support of the joint diagnoses (d1 ∪ d2 ) divided by the support of the condition diagnosis d2 . – The lift of the association rule L(d1 → d2 ) is the ratio between the confidence of the rule and the product of the support of both sides of the rule d1 and d2 . A lift greater than 1 means that diagnosis d1 is likely to be present in a prescription if diagnosis d2 is also present, while a lift less than 1 means that diagnosis d1 is unlikely to be present in a prescription where diagnosis d2 is present. A lift equal to 1 means that there is no association between diagnoses. These measures can be generalized to the case of multiple diagnoses; that is, the left side and/or the right side of the rule are itemsets of multiple diagnoses. As the confidence does not take into account the prevalence of d1 , confidence C(d1 → d2 ) can be divided by the support of d1 . Starting from this theoretical framework, our main purpose is to establish a link between association rules and network data, so that the transaction matrix T can be thought of as the affiliation matrix generated by a two-mode network or a bipartite graph B, defined above. In the case of medical prescriptions, we consider a prescriptions per diagnosis affiliation matrix as bipartite graph B = (D, P, A), where P is the set of nodes representing the prescriptions; D is the set of nodes representing the diagnoses, and the edge ai,j ∈ A ⊆ D × P is established between pj and di ; that is, an edge exists if and only if prescription pj contains diagnosis di . The collection of prescriptions per diagnoses, arranged in the binary affiliation matrix A, is considered a transaction matrix where each prescription is a transaction defined on the set P, so that D is the universal set of items, and each prescription in P is a subset of D. In this context, the aim stated in the association rules setting is to find sets of diagnoses that are strongly correlated in the prescription database, and are coherent with the notion of close neighbors in bipartite graphs; that is, pathologies directly connected to the same prescription. The common metrics for achieving this aim are the notions of support, confidence, and lift. These three measures induced by the association rules can be used to characterize the graph representation of

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the diagnosis itemset, defined as the projection of the bipartite graph induced by the prescription database. For the sake of computational efficiency, the analysis concentrates on the subset of diagnoses that have minimum support s. However, if the graph is partitioned in k groups, it makes sense to apply the association rules search on the separate subgraph induced by the partition. The more meaningful rules are usually sorted by the decreasing value of lift. The first important rules among diagnoses will help uncover associations between frequent patterns of co-occurrence in diagnoses, in the whole set of prescriptions.

4.2 Network Analysis Tools In the second step, being interested in the association of pathologies, we derived the one-mode network of the ICD-9-CM codes (diagnoses) by the association rules. The corresponding graph is represented by G (D, E, W), with D the set of ICD-9-CM codes, E ⊆ D ×D the set of edges, and W the set of weights. w : E → R; w(di , di  ) is the sum of the lift values of the rules with which the diagnoses are associated. To identify the most relevant and connected parts of the network that correspond to the most related pathologies, we used the k-core partitioning algorithm [10, 22]. Given a graph G (D, E, W), a k-core is defined as a subgraph H = (C, E|C) induced by the subset C ⊆ D; H is a k-core if ∀d ∈ C : degreeH (d) ≥ k. In addition, the subgraph H is the maximum subgraph with these characteristics. The k-core procedure is then used to extract the relatively dense subnetworks (i.e., the maximum density subgraph of the k-core), that is, the subset of kpathologies with the highest values of comorbidity occurrences, and to find cohesive subgroups of pathologies that are related by association rules with high lift. This network then is investigated with the usual exploratory tools of network analysis.

5 The Results First, to mine association rules, we reduced the original database to specific prescriptions and patients’ subpopulations as described below: – – – –

Type = DRUGs which are commonly related to actual diagnosis; AGE range, from 35 to 110 years; SEX, men and women, with two separate databases; and removing non-relevant ICD-9-CM codes (Pregnancy, Congenital, Newborn, Illdefined, etc.).

After the data were reduced, a set of 9845 patients (5252 women and 4593 men) were given 405,323 prescriptions (220,469 for women and 184,854 for men). The prescribed drugs totaled 627,924 (341,368 for males, and 286,556 for females). By

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applying the ICD-9-CM, we recognized a total number of 2387 diagnoses (1214 for women and 1713 for men), in a schematic view: P atients ⊂ P rescriptions ⊂ Drugs → Diagnoses : P atients × Diagnoses. Then, we set up a data matrix for men and one for women, in which the rows are patients, and the columns are diagnoses, to carry out separate analyses. Association rules were extracted according to the Apriori algorithm implemented in the R package arules [17] and visualized by arulesViz [16]. After a suitable setting of minimum support (0.01) and confidence (0.50), the algorithm extracted 517 rules for women and 175 rules for men plotted as scatterplots along their support and confidence, with the color intensity graded by lift in Fig. 2. Such graphical representation can be also interactively explored looking for more meaningful rules. An interactive data table visualization is given in Fig. 3 for the 10 most interesting rules. For the sake of simplicity, the results are decoded in Tables 3 and 4.

5.1 First Network Results Given the 517 association rules for women and the 175 corresponding rules for men, we derived two different one-mode networks. We apply the k-core partitioning algorithm to extract the relatively dense subnetworks, and to find cohesive subgroups of pathologies that are related by the derived association rules with high lift. Figure 4 portrays the two one-mode networks for women and men. The node is colored on the base of the k-core, while the node size is proportional to the betweenness centrality measures. Among the different network centrality measures [13], we selected the betweenness, because it helps in identifying the pathologies that are involved in many different rules, that is, in many comorbidity paths. In both figures, the red nodes correspond to the highest k-cores. It is possible to appreciate in both networks the diagnoses 401 and 462 are part of the central core, corresponding to hypertension and acute pharyngitis. Looking at the network for women, for example, the former is associated with some diseases that are directly connected, such as forms of atherosclerosis (414), bronchitis (491), artery occlusion stenosis (433), and hypertensive heart disease (402), and with some pathologies related to the aging process, such as arthrosis (715), and cataract (366), and to smoking, such as chronic bronchitis (491). In this network, an important node was given also by cystitis (595), which is also present in the network for men but with a lower betweenness centrality score. Looking at the network for men, for example, we notice that benign prostatic hyperplasia (600) appears in the highest k-cores.

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6 Discussion and Concluding Remarks Starting from the main findings, the first consideration involves the lift as an association rules’ measure, and its interpretation in the study of comorbidity. As stated in Sect. 2, in the contribution of Puzyrev [21] about the comorbidity phenomenon, two aspects of comorbidity are presented: (1) syntropy, or direct comorbidity, which is defined as mutual disposition, or the attraction of two or more diseases in the same individual; (2) dystrophy, or inverse comorbidity, which indicates those pathologies that are rarely found in the same patient at the same time. The lift appears to be a measure of these parameters, as a value greater than 1, meaning that diagnosis d1 is likely to be present in a prescription if diagnosis d2

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Fig. 3 Interactive data table visualization for the ten most interesting rules for women (a) and men (b)

is also present, thus, measuring syntropy. However, a value lower than 1 measures dystrophy, because the co-presence of the left-hand diagnosis and the right-hand diagnosis appears less often together than expected. In other words, the lift values could be used as a quantitative measure of comorbidity and an indicator of its quality characteristic. Another finding involves the sex effect. It clearly emerged that there are different association rules for men and women, as can be expected. Therefore, sex is a risk

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Table 3 Association rules for diagnoses—women 1 2 3 4 5 6 7 8 9 10

Left-hand rule → Right-hand rule Rhinitis due to pollen → Acute tracheitis Cervical spondylosis without myelopathy → Acute tracheitis {Acute bronchitis; Acute tracheitis} → Asthma {Esophagitis; Hypertensive heart diseas, benign} → Acute bronchitis {Esophagitis; Asthma} → Acute bronchitis {Acute bronchitis; Esophagitis} → Asthma {Esophagitis; Acute tracheitis} → Acute bronchitis {Asthma; Acute tracheitis} → Acute bronchitis {Acute tracheitis without mention of obstruction; Gastritis and duodenitis} → Osteoarthrosis, unspecified whether generalized or localized {Acute tracheitis; Hypertensive heart disease, benign} → Acute bronchitis

Table 4 Association rules for diagnoses—men 1 2 3 4 5 6 7 8 9 10

Left-hand rule → Right-hand rule Cervical spondylosis → Benign hypertensive heart disease; Cervical spondylosis → Acute tracheitis {Cystitis; Major depressive disorder, recurrent episode} → Hyperplasia of prostate {Osteoarthrosis and allied disorders; Cystitis; Essential hypertension} → Hyperplasia of prostate {Cystitis; Diseases of esophagus; Repair and plastic operations on joint structures} → Hyperplasia of prostate {Hyperplasia of prostate; Other forms of chronic ischemic heart disease; Diseases of the Respiratory System} → Cystitis {Periapical abscess; Hyperplasia of prostate; Essential hypertension } → Cystitis {Diabetes mellitus; Osteoarthros NOS-unspec} → Essential hypertension {Acute bronchitis and bronchiolitis; Diabetes mellitus} → Essential hypertension {Diabetes mellitus; Chr ischemic hrt dis NEC} → Essential hypertension

factor that strongly affects association rules mining. The extraction of many more association rules for men than women, as evidenced by comparing Fig. 2a and b, and the different network structures in Fig. 4a and b, confirms previous findings [6, 7] on the greater complexity of the clinical framework of women. In this sense, the concept of “gender medicine” [3] should be taken into account for complex system studies. It is focused on the differences in pathophysiology, clinical aspects, prevention, and treatment of diseases in the genders, and it will have a deep impact on health policy, research, and teaching. Further developments will include the assessment of age-class effects as a risk factor, considering its interaction with sex, too. Figure 4 has been reported in decoded form in Table 3 and Table 4, the entries are sorted by descending measure of Lift. The comparison of Tables 3 and 4 shows that there is a different complexity pattern of the ten most interesting rules between men and women, with the women (Table 3) showing a bigger presence of infectious disease related to respiratory and digestive system, and the men (Table 4) showing

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Fig. 4 One-mode networks for women (a) and men (b). Nodes are colored according to the k-core. Nodes’ size is set proportional to the betweenness centrality measure

a more complex pattern of interactions, with a limited presence of infectious diseases. These findings suggest that there is also a qualitative gender difference in comorbidity, although further studies are strongly advised to confirm the finding and exclude sources of bias. This difference indicates that probably one interactome1 is not enough, or at least, that we will need to understand how to navigate the

1 The interactome [26] is the whole set of molecular interactions in a given cell, or individual, and it is represented as a graph of the biological network. It includes all the interactions between molecules belonging to different biochemical families, such as nucleic acids, proteins, lipids, carbohydrates, hormones, etc. Interactomics [18] is a discipline at the intersection of bio-informatics and biology that deals with the study of networks of interactions and their consequences.

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interactome in different ways, according to the gender of the subjects under study. According to the findings of the present study, interactomics will probably need to add the dimension of gender differences to the field of study. As future lines of research, the temporal dimension could be added to study the evolution of rules in the same cohort of patients over time. For instance, the lift metric could become an instrument of computational epidemiology, building software that automatically extracts data from a large database of EHRs. Such a monitoring instrument may measure the presence and variation of syntropy and dystrophy by gender and age, obtaining the trajectories of comorbidity in a given population. The trajectories, moreover, may be studied by plotting the value of lift for a given association in a cohort of patients followed during a certain number of years. The lift data could also be used for tracking population epidemiology relating to geographic, environmental, social, cultural, and other parameters, using a big data approach. A translational application could be a sort of computational epidemiology monitoring system. Finally, studies should also be performed using targeted data extraction approaches of EHRs, but this needs a much larger dataset. The more specific the analysis, the less numerous becomes the sample, and thus, the data extraction process strongly reduces the data yield.

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12. Feinstein, A.R.: The pre-therapeutic classification of co-morbidity in chronic disease. J. Chronic Dis. 27, 455–468 (1970) 13. Freeman, L.C.: Centrality in social networks conceptual clarification. Soc. Netw. 1(3), 215–239 (1978) 14. Gamberger, D., Lavrac, N., Jovanoski, V.: High confidence association rules for medical diagnosis. In: Proceedings of IDAMAP99, pp. 42–51 (1999) 15. Gupta, N., Mangal, N., Tiwari, K., Mitra, P.: Mining quantitative association rules in protein sequences. In: Williams, G.J., Simoff, S.J. (eds.) Data Mining, LNAI 3755, pp. 273–281. Springer, Berlin (2006) 16. Hahsler, M.: arulesViz: interactive visualization of association rules with R. R J. 9(2), 163–175 (2017) 17. Hahsler, M., Chelluboina, S., Hornik, K., Buchta, C.: The arules R-package ecosystem: analyzing interesting patterns from large transaction datasets. J. Mach. Learn. Res. 12, 1977– 1981 (2011) 18. Kiemer, L., Cesareni, G.: Comparative interactomics: comparing apples and pears? Trends Biotechnol. 25(10), 448–454 (2007) 19. Mercer, S.W., Smith, S.M., Wyke, S., O’dowd, T., Watt, G.C.: Multimorbidity in primary care: developing the research agenda. Fam. Pract. 26, 79–80 (2009). Available via DIALOG. https:// academic.oup.com/fampra/article/26/2/79/2367540. Cited 07 May 2018 20. Pfaundler, M., von Seht, L.: Uber Syntropie von Krankheitszustanden. Z. Kinderheilk. 30, 298–313 (1921) 21. Puzyrev, V.P.: Genetic bases of human comorbidity. Genetika 51, 491–502 (2015) 22. Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5(3), 269–287 (1983) 23. Serban, G., Czibula, I.G., Campan, A.: A programming interface for medical diagnosis prediction. Stud. Univ. Babes-Bolyai Inform. LI, 21–30 (2006) 24. Valderas, J.M.: Increasing clinical, community, and patient-centered health research. J. Comorb. 3, 41–44 (2013) 25. Valderas, J.M., Starfield, B., Sibbald, B., Salisbury, C., Roland, M.: Defining comorbidity: implications for understanding health and health services. Ann. Fam. Med. 7, 357–363 (2009) 26. Vidal, M., Cusick, M.E., Barabási, A.L.: Interactome networks and human disease. Cell 144(6), 986–998 (2011) 27. World Health Organization: International classification of Diseases (ICD) (2010). Available from: http://www.who.int/classifications/icd/en/ 28. Yurkovich, M., Avina-Zubieta, J.A., Thomas, J., Gorenchtein, M., Lacaille, D.: A systematic review identifies valid co-morbidity indices derived from administrative health data. J. Clin. Epidemiol. 68, 3–14 (2015)

A Mixture Model Approach for Clustering Bipartite Networks Isabella Gollini

Abstract This chapter investigates the latent structure of bipartite networks via a model-based clustering approach which is able to capture both latent groups of sending nodes and latent variability of the propensity of sending nodes to create links with receiving nodes within each group. This modelling approach is very flexible and can be estimated by using fast inferential approaches such as variational inference. We apply this model to the analysis of a terrorist network in order to identify the main latent groups of terrorists and their latent trait scores based on their attendance to some events.

1 Introduction In recent years, there has been a growing interest in the analysis of network data. Network models have been successfully applied to many different research areas. We refer to [1] for a general overview of the statistical models and methods for networks. In this chapter we will focus on finding clusters in a particular class of networks that is called bipartite networks. Bipartite networks consist of nodes belonging to two disjoint and independent sets, called sending and receiving nodes, such that every edge can only connect a sending node (e.g., actor) to a receiving node (e.g., event). Latent variable models have been used to model the unobserved group structure of bipartite networks by setting sending nodes as observations and receiving nodes as observed variables (see, for example, [2, 3]). One important issue limiting the use of classical latent variable approaches, such as latent class analysis [4, 5] and stochastic blockmodels [6], is the assumption of local independence within the groups that, in the presence of a large heterogeneous network, may tend to yield

I. Gollini () University College Dublin, Belfield, Dublin, Ireland e-mail: [email protected] © Springer Nature Switzerland AG 2020 G. Ragozini, M. P. Vitale (eds.), Challenges in Social Network Research, Lecture Notes in Social Networks, https://doi.org/10.1007/978-3-030-31463-7_6

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an overestimated number of groups making the results more difficult to interpret and potentially misleading. Aitkin et al. [4] proposed to use different models to overcome the issue of the local dependence assumption including the random Rasch latent class model in which they made use of class and event specific parameters that are, however, not able to capture the within class behaviour of each actor. Furthermore the computational effort required to estimate the model they propose is significant and this issue makes inference infeasible for large networks. This chapter concerns the identification of groups in bipartite networks consisting of a set of actors and a set of events through a statistical mixture modelling approach which assumes the existence of a latent trait describing the dependence structure between events within actor groups and therefore capturing the heterogeneity of actors’ behaviour within groups. This modelling framework allows for: model selection procedures for estimating the number of groups; explanation of the dependence structure of events in each group; description of the behaviour of each actor within each group by quantifying, through the latent trait, the conditional probability that a certain actor belonging to a certain group will attend a certain event. The posterior estimate of the latent trait scores can be visualized so as to interpret the estimated latent traits within each group. In order to fit the model variational inferential approaches are applied (see [7] and [8] for a comparison of estimates given by the variational and other approaches in latent trait models). The code implemented is included in the lvm4net package [9] for R [10]. The rest of this chapter is organized as follows: in Sect. 2 we describe the model and the inferential approach. In Sect. 3 we apply the proposed methodology to the Noordin Top terrorist bipartite network [4] in which we will aim to identify clusters of terrorists based on their attendance to a series of events in Indonesia from 2001 and 2010. We conclude in Sect. 4 with some final remarks.

2 Model-Based Clustering for Bipartite Networks The relational structure of a bipartite network graph can be described by a random incidence matrix Y on N sending nodes (i.e., actors), R receiving nodes (i.e., events) and a set of edges {Ynr : n = 1, . . . , N ; r = 1, . . . , R}, where Ynr =

 1, n ∼ r; 0, n ∼  r.

To cluster bipartite networks we adapt a flexible model-based clustering approach for categorical data, the mixture of latent trait analysers (MLTA) model introduced by Gollini and Murphy [8], to the context of bipartite network data. The MLTA model is a mixture model for binary data where observations are not necessarily conditionally independent given the group memberships. In fact, the observations within groups are modelled using a latent trait analysis model and thus dependence is accommodated. The MLTA model generalizes the latent class analysis and latent

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trait analysis by assuming that a set of N sending nodes can be partitioned into G groups, and the propensity of each actor to create links to the R receiving nodes depends on both the group they belong to and the presence of a D-dimensional continuous latent variable θ n . The model assumes that each sending node comes from one of G unobserved groups and defines zn = (zn1 , zn2 , . . . , znG ) as an indicator of the group membership, zng = 1 if actor n is from group g, with the following distribution: zn ∼ Multinomial(1, (η1 , η2 , . . . , ηG )), where ηg is the prior probability of a randomly chosen observation coming from  group g ( G g  =1 ηg  = 1 and ηg ≥ 0 ∀ g = 1, . . . , G). Further, the conditional distribution of yn1 , . . . , ynR given that the observation is from group g is assumed to be a latent trait model with parameters brg and wrg . Thus, the likelihood of the MLTA model is defined as p(y) =

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R   y  1−ynr πrg (θ n ) nr 1 − πrg (θ n ) , r=1

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1  , 1 + exp −(brg + wTrg θ n )

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In addition, it is assumed that the D-dimensional latent variable θ n ∼ N(0, I). The attractiveness of receiving node r for sending nodes belonging to group g is modelled by the parameter brg . The parameter wrg measures the heterogeneity of the behaviour of sending nodes belonging to group g to connect to the receiving

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node r (i.e., the heterogeneity of terrorists belonging to the latent group g in attending event r); it also accounts for the dependence between receiving nodes. The vector θ n contains the latent variables explaining the propensity of forming links for sending node n, i.e., the propensity of terrorist n to attend the events. We also use a constrained model with common variable-specific slope parameters across groups (i.e., wrg = wrg  = wr , where g = g  ): πrg (θ n ) =

1

, 1 + exp −(brg + wTr θ n )

0 ≤ πgr (θ n ) ≤ 1,

This model is particularly useful to avoid the estimation of too many parameters, especially when the data set is complex, with actors coming from several latent groups and the continuous latent variable having high dimensionality. The likelihood of the MLTA model is computationally intractable. For this reason [8] proposed to use a double EM algorithm with variational approximation of the likelihood to fit this model, also guaranteeing fast convergence. The main aim of this variational approach is to maximize the Jaakkola and Jordan [11] lower bound of the likelihood function. This lower bound is a function of auxiliary parameters, called variational parameters, that are optimized to tighten this lower bound. The standard errors of the model parameters can be calculated using the jackknife method [12]. For full details of the double EM algorithm, we refer to [8]. Since the EM approach is adopted, there is the issue that the results may be affected by the risk of converging to a local maximum instead of the global maximum approximate likelihood. For this reason, it is generally advisable to run the algorithm several times using different initializing values, and select the solution with maximum approximate likelihood. The application of the variational approach makes the estimation procedure much more efficient than most of the classical simulation-based estimation methods even when multiple starts are employed. However, the approximation of the log-likelihood obtained by using the variational approach with the Jaakkola and Jordan lower bound is always less or equal than the true log-likelihood, so before performing model selection based on the likelihood, like the Bayesian Information Criterion (BIC) [13], it may be advantageous to get a more accurate estimate of the log-likelihood at the last step of the algorithm using Gauss–Hermite quadrature [8].

3 Noordin Top Terrorist Network The Noordin top terrorist network data [14] displayed in Fig. 1 is a bipartite network oriented around the Malaysian Muslim extremist Noordin Mohammad Top (ID: 54) and his collaborators (the data set is available in the manet package [15] for R). The data include relational information on N = 79 sending nodes that are individuals belonging to terrorist/insurgent organizations and on R = 45 receiving nodes that

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represent events in Indonesia and nearby areas from 2001 to 2010. The incidence matrix contains links encoding the attendance behaviour of the terrorists to the events.

3.1 Statistical Analysis We apply the MLTA modelling approach to the Noordin Top Terrorist Network. To avoid the issue of getting estimates affected by convergence to a local maximum, we use ten random starts of the algorithm and only the estimates corresponding to the maximum likelihood value are selected. The model parameters brg and wrg are initialized by random generated numbers from a N(0, I) and the variational parameters are initialized to be equal to 20 in order to reduce the dependence of the final estimates on the initializing values.

84 Table 1 BIC results for standard and constrained MLTA models with different number of groups and dimensions

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G=2 G=3 G=4

D=0 2062 2157 2290

D=1 Common wr 2138 2034 2403 2115 2730 2249

D=2 Common wr 2389 2096 2876 2229 3385 2385

D=3 Common wr 2793 2311 3417 2434 4419 2595

Value in bold denotes the smallest BIC value

The model is fitted on a range of groups, from 2 to 4 and the continuous latent variable takes value D from 0 to 3. For D = 0 the MLTA model reduces to a latent class analysis where the observations are assumed to be conditionally independent given the group membership. Model selection is performed on both the unconstrained MLTA and the constrained model with common slope. The Bayesian Information Criterion (BIC) [13] is used to select the best model, and it is defined as: BIC = −2GH + k log(N ), where GH is the estimate of the log-likelihood at the last step of the algorithm obtained by using Gauss–Hermite quadrature, k is the number of free parameters in the model and N is the number of sending nodes. The model with the lower value of BIC is preferable. Table 1 shows the BIC values for models with increasing dimensionality. The best model selected is the one with two groups, a one-dimensional latent trait and common slope across groups. For the best model selected, the values of the mixing proportions are: η1 = 0.57 (SE = 0.080) for Group 1, and η2 = 0.43 (SE = 0.084) for Group 2.

3.2 Interpreting the Actor’s Behaviour The sending nodes are partitioned into the two groups according to their maximum a posteriori (MAP) probability that they belong to each group. Figure 2 shows the posterior probability of each actor to belong to each group. Most of the terrorists have been assigned to a particular group with probability very close to 1. In particular, Noordin Top (ID 54), attending 23 events, and Azhari Husin (ID 21), attending 17 events, are allocated together into Group 1 with probability 1. The ‘lone wolves’ (IDs 75, 76, 77, 78, 79), i.e., terrorists who have not attended any event, have been assigned to Group 1, but the uncertainty associated with their group membership is very large: in fact, their posterior probability to belong to Group 1 is 0.6. In order to have a deeper understanding of group memberships we can use the information provided by the posterior distribution of the latent trait score θn conditional on the observation belonging to a particular group which can be obtained from the model estimates (see Fig. 3).

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Probability to belong to group 1 1 3 6 12 15 16 18 19 20 21 24 25 26 31 32 42 10 5274 38 44 45 47 46 48 49 50 51 54 55 56 59 60 62 64 66 67 70

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Fig. 2 Probability to belong to a group for each terrorist in the best model selected (two groups, a one-dimensional latent trait and common slope across groups)

The posterior mean estimates of these θn together with the information about event attendance ynr can be used to interpret the latent variables within each group: Fig. 4 allows us to notice that in Group 1 the terrorists with high values went to events 7, 14 and 22 and most of the terrorists with low values went to events 13, 26, 34, 42. In Group 2 positive values are assigned to those terrorists who attended event 1 (it is also possible to notice that none of the terrorist in Group 1 attended event 1), negative values of the latent trait are associated with terrorists who attended events 2, 9, 25 and 33.

3.3 Interpreting the Events Attendance A measure of the heterogeneity of attending event r within group g is given by the slope value wrg ; the larger the value of wrg , the greater the differences in the probabilities of sending a link (going to event) r for actors from group g. The choice of a model with the common slope (wr1 = wr2 = wr ) in all groups suggests the latent trait has the same effect in all groups. From Fig. 5 it is possible to notice that most of the slope parameters are non-zero, meaning that the latent trait introduces significant variation within the groups. This indicates that there is

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21

Probability 0.5

0.75

1

Group 2

Fig. 3 Latent structure of the bipartite network. The terrorists are partitioned into the two groups according to their maximum a posteriori (MAP) probability and plotted according to their latent trait position. The darker the vertex colour, the higher their MAP of belonging to the group

considerable variability within the event attendance in the two groups and that some events are positively dependent (i.e., those going/not going to one event will tend to be going/not going in the others events) and others are negatively dependent. The dependence between events r and k in group g is given by wTrg wkg , and the results are shown in Fig. 6. Red (blue) squares in the heatmap mean positive (negative) dependence, the darker they are, the higher is the dependence between two events. Figure 6 shows that the two set of events (1, 6, 7) and (3, 33, 36) are positively dependent within them and negatively between them. The heatmap displayed in Fig. 7 represents the values of the log{lift} [16] that can be used to quantify within each group the effect of the dependence on the probability of attending two events compared to the probability of attending two events under an independence model. Mathematically the log{lift} for events r and k for actors belonging to group g is defined as 

  P ynr = 1, ynk = 1|zng = 1   , log{lift} = log P ynr = 1|zng = 1 P ynk = 1|zng = 1 where r = 1, 2, . . . , R and r = k. Two independent events have log{lift} = 0: the more two events are positively dependent, the higher the value of the log{lift}. Lift

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Event

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Fig. 4 Posterior mean estimate of the latent trait scores θn within each group for each actor and attendance to event

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Fig. 5 Estimates of slope parameters for each receiving node (event) in the network and associated 95% confidence interval

values that are much less than 0 provide evidence of negative dependence within groups. Figure 7 shows that in Group 1 there is high negative dependence between events 1 and events 3, 33, 36, and in Group 2 there is high positive dependence between the events 27, 32, 34, 35, 45.

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44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

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Fig. 6 Dependence between events, calculated as wTr wk

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44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Fig. 7 Log-Lift for each pair of receiving nodes (events) of the network

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Fig. 8 Estimates of the intercept parameters for each receiving node (event) and corresponding 95% confidence intervals

The attractiveness of event r for actors belonging to group g is modelled by brg . Figure 8 shows that most of the values are significantly negative highlighting the sparse structure of the network. Since θn ∼ N(0, 1), the probability that the median individual in group g attends events r can be calculated from the attractiveness parameters through the relationship: πrg (0) = p(xnr = 1|θn = 0, zng = 1) =

1 . 1 + exp(−brg )

From Fig. 9 it is evident the different behaviour of the actors belonging to the two groups. Actors in Group 1 have high probability to attend events 7, 13, 14, 43, 44, 45, while those in Group 2 have very low probability to attend these events ( 0 and ΔUji (DDEA − ji , Xj , Xi , εji ) = 0; • For each DDEAij = 0, ΔUij (DDEA − ij , Xi , Xj , εij ) = 0 =⇒ ΔUji (DDEA − ji , Xj , Xi , εji ) = 0. So, a possible choice can be for example     DDEAij =1 Uij DDEA−ij , Xi , Xj , εij >0,Uj i DDEAij , Xj , Xi εj i =0 ∀i=j (5) and assuming εij = 0 we will have     DDEAij = 1 Uij DDEA−ij , Xi, Xj > 0, Uj i DDEAij , Xj , Xi = 0 ∀i = j (5a) The choice of a DDEAij link in (5) depends on the choice of other DDEA − ij . This indicates that we cannot treat each linkas a single observation and use a dyadic regression because DDEAij is endogenous in the model because it can be related to (εij , εji ). But since ΔUji = 0 the choice is only for the agent i with εij . What futher complicates the statistical inference of (5) is that there are multiple equilibrium, which will influence the identification of the parameters (but this is not done here and may be of interest for further work developments). In (5a), however, we can assume independence of choice (because εij = 0). At the same time we do not have reciprocity because ΔUji (the marginal utility of the agent j in the link ij) will always be zero and the agent j will have no incentive to form some kind of link with i. Network Formation Mechanism (and a Hint at Econometric Specification) DMUs simultaneously announce the desired output link (we remind that the DMU i-th unilaterally decides on the formation of the link ij according to the indications of the analysis DEA of peers) and this happens under the assumption of complete information. So each DMU i gets a payoff as in (3) under the pairwise stability in (5a). Once the mechanism is specified the DDEA network is a strategy in Nash equilibrium if it resolves the following system of N (N − 1) equations:   DDEAij = 1 θ Xij + εij ≥ 0

∀i = j with i, j ∈ DDEAij

(6)

A graphical example of simultaneity is shown in Fig. 2. The inefficient agents with U > 0 will announce the desired output link according to the strategy defined by the peer group analysis as developed in Appendix 1. For example following the example of Appendix 1 the unit C will have a strategy training of two links with agents E and F. In summary we will say that “in this DEA-

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Fig. 2 Simultaneous mechanism formation of DEA-based network Table 1 Distribution of net utility under cij = 0 DMU 1 2 3 4 5 6

Formula (7) (1 − 1) × 0 = 0 (1 − 0.858) × 0.322 = 0.0457 (1 − 0.789) × 0.434 = 0.092 (1 − 0.828) × 0.596 = 0.102 (1 − 1) × 0 = 0 (1 − 1) × 0 = 0

– (1 − 0.858) × 0.677 = 0.096 (1 − 0.789) × 0.565 = 0.119 (1 − 0.8281) × 0.403 = 0.069

based network version agents will strategically form links with other agents if the marginal utility is positive U > 0 and with probability of λ·ij independently of other links” [36] shall apply. Example Suppose that there are N = 6 DMUs with the same technology (2 inputs and 1 output) and suppose that once applied the DEA (model (11) in Appendix 1) the DEA score efficiency for each of it are the following: 1, 0.858, 0.789, 0.828, 1, 1 (at this time the network is empty, yet). We now assume a further specification of the utility function (3) (with εij = 0) as follows: Ui =

1−N j

(1 − DEAeffscore) λij · − cij

(7)

where λ·ij are the optimal lambda and cij i costs to maintain and form links.Suppose we then have a lambda matrix like the one in Appendix 2. Assuming cij = 0 we will have the following net utility distribution (Table 1):

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In the empty network we will have Ui = 0, always. The marginal utility U for DMU 3 will be distributed for example as follows: 3 → 5:0.092 and 3 → 6:0.119. That is, unit 3 will have a marginal utility of 0.119 in forming the link with unit 6 and a marginal utility of 0.092 in forming the link with unit 5 (so that the sum will be just equal to 1–0.789 = 0.211 (0.092 + 0.119)). In the end, all strategies have been completed (simultaneously) resulting in the same network as right picture in Fig. 2. Statistical Point of View Conditional on (5a) the DEA-based network is based on independent choices to only send links to the outside. Statistically speaking this allows us to focus on Dij = (Xij ) dyads with only these states : (1) Dij = 0 dyad null (if U = 0 for the agent i and U = 0 for agent j), and (2) Dij = 1 dyad asymmetric (i·j) (if U > 0 for agent i and U = 0 for agent j). In the case of reciprocity it would mean that the efficient agent also sends a link to the inefficient one. But this would contradict the above economic model as it would contradict the “lesson of learning from the best” by the inefficient agent. The link resulting from the agent’s strategic decision can be considered as a random variable with some type of distribution and therefore the adjacency matrix in Appendix 2 considered as a possible realization of this random variable shall be used. Under the assumption of independence the links between the nodes follow a distribution of Bernoulli with equal probability, say p. And as we know for n→ is a small p the distribution of degrees will follow a Poisson and then we will have a random graph model [25, 41]. However this is not our case where by placing pij (1, 0), the probability of sending a link by agent i to the agent j, and pij (0, 0), the probability of null dyad following [26], we can write the following simple statistical model: log pij (0, 0) = λij

and

log pij (1, 0) = λij + αi + θ

(8)

λij , α i , θ are the parameters of the model. α i controls the effect of a link leaving i, θ is the general propensity of the DEA network to have a link and λij is a normalization constant that ensures (2) to maintain for each dyad (i, j). pij (0, 0) + pij (1, 0) = 1

(9)

Thus the model (8) does not include any in-link effects or/and reciprocity. The outgoing link is now strategically based. Network Effects Modeling the DEA-based network under the assumption of independence means that the agent’s behavior does not depend on the relationships and actions of others in the rest of the network. In other words from this assumption follows the absence of network effects. Assuming that each individual has preference given by (7) the utility of agents when they take action is (1 − DEAeffscore) * λij and there is the cost of this action. Thus agents take actions α i = 1 if (1 − DEAeffscore) λij > c and do not take actions α i = 0 if (1 − DEAeffscore) λij ≤ c. Assuming that c = 0 means that in the absence of network effects a proportion of agents for which (1 − DEAeffscore) λij > 0 will take

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action for which DEAeffscore* λij < 1. In other words equilibrium in the absence of network effects only inefficient agents will take action.

4 Micro and Macro Analysis Suppose you have a sample of N = 40 units using the same technology (two inputs and an output) and have a DEA-based network. The questions for a researcher can be: (1) the network is dense? (2) how many connections exist on average in the network? (3) is there any unit that forms more ties than the others? the links have the same structures? And so on.

4.1 Macro Analysis Our macro analysis will consider: (1) network density, (2) average outdegree, and (3) average distance and diameter. In simple patterns of network formation, the distribution of Poisson emerges when the links are formed uniformly at random (and are also dense) so that the difference in grade between nodes reflects causality relative to a binomial variable. A distribution exhibiting the law of power instead has a large variation in degrees and is usually derived from a process “rich seeks rich”: a dynamic in which the nodes with a higher grade are the nodes gaining new ties at a higher rate [42, 43]. Differences in grade distribution have an important implication for diffusion processes. The average degree between nodes equals the number of ties along the shortest route between nodes. The average distance between them has important consequences from the economic point of view. The fact that social networks have an average short distance is the characteristic studied by Milgram [44]. Generally route lengths tend to vary with the logarithm of the number of nodes in a network. Theorems that have presented that this is true have been tested for many network models, starting with some simple ones such as [25, 45]. The diameters tend to be small in random graphs for the reasons mentioned above, while it is for many different reasons that the diameter is short for networks in which individuals choose links strategically. The diameters are directly useful to offer limits for some diffusion processes that travel through shorter paths [46]. As we can see our network of 40 DEA-based virtual enterprises forms just 6% of all possible links (this measure considers all possible links without distinguishing between in and out), and on average each sender (agent proposing the link) sends 2 proposals for links (see Table 2). Reaching every other agent is relatively easy (average diameter is 1). Very interesting in this is that our network is the distribution of the out and in degree. The second is very concentrated with a coefficient of Gini equal to 0.953 (see Table 2), while the first is relatively uniform with a coefficient of Gini equal to 0.211 (see also the Lorenz curves in Fig. 1). Our interpretation of Gini’s coefficient for the outdegree, in this example, is that the probability for a node

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Table 2 Macro level analysis

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Value 0.058 2.275 2.275 1 0.211 0.953 0.966

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Fig. 3 Lorenz curve for in/out degree Table 3 Triad census

Type of triad 003 012 102 021D 021U All others

Frequency 7831 640 0 77 1332 0

to have some outdegree is about the same (under the assumption of independent choices). The Gini coefficient for indegree confirms the concentration of sender choices on a few items as we can see in Fig. 3. Our macro analysis concludes with structural analyses and may concern the census of the dyads and the triads, here we will report only the analysis of the triads as in Table 3. Triad 003 means that there are no mutual links (mutuality = 0), that there are no asymmetric links (asymmetry = 0), but that there are three null dyads in the triad (nullity = 3). The interpretations of the triads in the case of DEA-based network are, in our opinion, the following: (1) 003 means empty triad containing agents of the same type (or all efficient or all inefficient) or a combination of them, (2) 012 means that the triad contain one asymmetrical link and two null dyads. This means that in the DEA-based network this triad can contain one inefficient and two efficient

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agents, only, (3) 021 U means that the triad contain two asymmetrical dyads and one null dyad. This means that it can contain two inefficient agents and one efficient agent only, (4) 021D means triad can contain two asymmetric dyads and one null dyad. This triad is similar to the last one (i.e. 021U) but can contain one inefficient agent and two efficient ones, only finally (5) we observe the triad 021C is illogical in a DEA-based network as this triad would assume the existence of a link from the efficient agent to the inefficient one, that is a j → i link. All remaining triads with 1 and 2 as the first element (for example 111D, or 201, and so on) cannot (and should not) exist in a DEA-based network because they presuppose a mutual link of the type i → j and j → j. As well as triads of type 030T and 030C assuming transitivity or cycles in them cannot exist in a network just on the DEA.

4.2 Micro Analysis Micro level analysis concerns the centrality and structural equivalence. There are numerous measures of centrality [47–49] and they can affect concepts such as influence, prestige, popularity, and so on [50, 51]. These last measures have implications for the processes of diffusion within networks, or the impact of a node on transactions [46]. Two nodes are structural equivalents if they have the same relationship with all other nodes [49, 52, 53]. The phenomena which have taken this measure into account include the evolution of inter-organizational networks (i.e., [54]) or the diffusion of innovation (i.e., [55]) (Table 4). For the DEA-based network we will say that two nodes are structurally equivalent if they send the same links to the same efficient agents (see Fig. 4 below). Table 4 Indegree of DEA-based network of 40 virtual firms

Fig. 4 Structural equivalence example

DEA full efficient actor 8 13 21 24 34

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5 Estimation and Simulation As presented above we are assuming our statistical model (8) belongs to the family of exponential random graph models [26]. That said, we place the following loglinear specification:   

 log Pθ , y Y = yij = φij yij − log κ (θ, y)

(10)

i=j

where φ ij = α i + θ and α i are the sender’s effect and the general effect of the links similar to the interception of a linear regression. The specification in (10) differs from the model p1 [26] because no transitivity effect is considered and the term φ ij considers only the effect of the sender. The (10) model can be estimated using the estimation method of approximate maximum likelihood estimates obtained using a stochastic algorithm based on the Markov Chain Monte Carlo (MCMC). Obviously, in our intention model (10) should generate a network with the same structural proprieties of a DEA-based network. The estimates are given in Table 5 below. From Table 5 we observe a positive sender effect for inefficient agents and a negative effect for efficient ones (8,13,21,24,34 are DEA efficient nodes). The expected probability of observing a direct link from the inefficient agent sender in the DEA network to the efficient one in the current network is increasing and statistically significant and the expected probability of observing a direct link from the efficient agent never sender in the DEA-based network to the inefficient one is decreasing and statistically significant.

5.1 Simulation In this section: (1) we will simulate different statistical models assumed as benchmarks and compare them with our network, (2) we simulate DEA-based networks of different sizes and (3) we simulate net from our ERGM model and compare it

Table 5 ERGM estimates Sender (2,4,5,6,9,11,20,22,25,27,30,32,35) Sender (3,7,10,12,14,15,16,17,18,19,23,26,28,29,36,37, 38,39,40) Sender (8,13,21,24,34) Sender (33) Node.Factor = Status.Ineff AIC *** Significance at 0.001%

Estimates 6.0404 6.8186

Std. error 1.1713 1.1689

Pr (>|z|)